×

Convex geometry and its applications. Abstracts from the workshop held December 12–18, 2021 (hybrid meeting). (English) Zbl 1506.00039

Summary: The geometry of convex domains in Euclidean space plays a central role in several branches of mathematics: functional and harmonic analysis, the theory of PDE, linear programming and, increasingly, in the study of algorithms in computer science. The purpose of this meeting was to bring together researchers from the analytic, geometric and probabilistic groups who have contributed to these developments.

MSC:

00B05 Collections of abstracts of lectures
00B25 Proceedings of conferences of miscellaneous specific interest
52-06 Proceedings, conferences, collections, etc. pertaining to convex and discrete geometry
52Axx General convexity
68Q25 Analysis of algorithms and problem complexity
60D05 Geometric probability and stochastic geometry
Full Text: DOI

References:

[1] S. Artstein-Avidan, D. Florentin, and Y. Ostrover, Remarks about mixed discriminants and volumes, Commun. Contemp. Math., 2 (2014), 1350031. · Zbl 1292.52004
[2] S. Bobkov and M. Madiman, Reverse Brunn-Minkowski and reverse entropy power inequal-ities for convex measures, J. Funct. Anal., 262 (2012), 3309-3339. · Zbl 1246.52012
[3] W. Fenchel, Generalisation du theoreme de Brunn et Minkowski concernant les corps con-vexes, C. R. Acad. Sci. Paris. 203 (1936), 764-766. · Zbl 0015.12007
[4] M. Fradelizi, A. Giannopoulos, and M. Meyer, Some inequalities about mixed volumes, Israel J. Math., 135 (2003), 157-179. · Zbl 1045.52002
[5] I. Soprunov, and A. Zvavitch, Bezout inequality for mixed volumes, International Mathe-matics Research Notices. IMRN, 23 (2016), 7230-7252. · Zbl 1404.52008
[6] Jian Xiao Bézout-type inequality in convex geometry, International Mathematics Research Notices. IMRN, 16 (2019), 4950-4965. · Zbl 1462.52011
[7] Gautier Berck, Andreas Bernig, and Constantin Vernicos, Volume entropy of Hilbert ge-ometries, Pacific J. Math. 245 (2010), no. 2, 201-225, DOI 10.2140/pjm.2010.245.201. MR2608435 · Zbl 1204.52003 · doi:10.2140/pjm.2010.245.201.MR2608435
[8] Jean-Luc Brylinski, The beta function of a knot, Internat. J. Math. 10 (1999), no. 4, 415-423, DOI 10.1142/S0129167X99000161. MR1697615 · Zbl 0972.57006 · doi:10.1142/S0129167X99000161.MR1697615
[9] Bruno Colbois and Patrick Verovic, Two properties of volume growth entropy in Hilbert ge-ometry, Geom. Dedicata 173 (2014), 163-175, DOI 10.1007/s10711-013-9934-2. MR3275296 · Zbl 1305.53028 · doi:10.1007/s10711-013-9934-2.MR3275296
[10] Dmitry Faifman, A Funk perspective on billiards, projective geometry, and Mahler volume, available at arXiv:2012.12159[math.DG].
[11] Michael H. Freedman, Zheng-Xu He, and Zhenghan Wang, Möbius energy of knots and unknots, Ann. of Math. (2) 139 (1994), no. 1, 1-50, DOI 10.2307/2946626. MR1259363 · Zbl 0817.57011 · doi:10.2307/2946626.MR1259363
[12] Y. Gordon, M. Meyer, and S. Reisner, Zonoids with minimal volume-product-a new proof, Proc. Amer. Math. Soc. 104 (1988), no. 1, 273-276, DOI 10.2307/2047501. MR958082 · Zbl 0663.52003 · doi:10.2307/2047501.MR958082
[13] Hiroshi Iriyeh and Masataka Shibata, Symmetric Mahler’s conjecture for the volume product in the 3-dimensional case, Duke Math. J. 169 (2020), no. 6, 1077-1134, DOI 10.1215/00127094-2019-0072. MR4085078 · Zbl 1439.52007 · doi:10.1215/00127094-2019-0072.MR4085078
[14] Gil Kalai, The number of faces of centrally-symmetric polytopes, Graphs Combin. 5 (1989), no. 1, 389-391, DOI 10.1007/BF01788696. MR1554357 · Zbl 1168.52303 · doi:10.1007/BF01788696.MR1554357
[15] Erwin Lutwak, Extended affine surface area, Adv. Math. 85 (1991), no. 1, 39-68, DOI 10.1016/0001-8708(91)90049-D. MR1087796 · Zbl 0727.53016 · doi:10.1016/0001-8708(91)90049-D.MR1087796
[16] Jun O’Hara, Energy of a knot, Topology 30 (1991), no. 2, 241-247, DOI 10.1016/0040-9383(91)90010-2. MR1098918 · Zbl 0733.57005 · doi:10.1016/0040-9383(91)90010-2.MR1098918
[17] Athanase Papadopoulos and Marc Troyanov (eds.), Handbook of Hilbert geometry, IRMA Lectures in Mathematics and Theoretical Physics, vol. 22, European Mathematical Society (EMS), Zürich, 2014. MR3309067 · Zbl 1310.51001
[18] J. Saint-Raymond, Sur le volume des corps convexes symétriques, Initiation Seminar on Analysis: G. Choquet-M. Rogalski-J. Saint-Raymond, 20th Year: 1980/1981, Publ. Math. · Zbl 0531.52006
[19] Univ. Pierre et Marie Curie, vol. 46, Univ. Paris VI, Paris, 1981, pp. Exp. No. 11, 25 (French). MR670798
[20] Nicolas Tholozan, Volume entropy of Hilbert metrics and length spectrum of Hitchin representations into PSL(3, R), Duke Math. J. 166 (2017), no. 7, 1377-1403, DOI 10.1215/00127094-00000010X. MR3649358 · Zbl 1375.53022 · doi:10.1215/00127094-00000010X.MR3649358
[21] Constantine Vernicos and Cormac Walsh, Flag-approximability of convex bodies and vol-ume growth of Hilbert geometries, Annales de l’ens, in press, available at arXiv:1809. 09471[math.MG].
[22] Gilles Bonnet, Daniel Dadush, Uri Grupel, Sophie Huiberts, Galyna Livshyts, Asymptotic Bounds on the Combinatorial Diameter of Random Polytopes, under submission.
[23] Karl Heinz Borgwardt, The Simplex Method: a probabilistic analysis, Springer, 1987. References · Zbl 0604.90092
[24] S. Chatterjee, Superconcentration and related topics. Springer Monographs in Mathematics. Springer, Cham, 2014. · Zbl 1288.60001
[25] W. Chen, M. Handschy and G. Lerman On the energy landscape of the mixed even p-spin model. Probab. Theory Related Fields 171 (2018), no. 1-2, 53-95. · Zbl 1417.60079
[26] A. Baernstein II, Symmetrization in Analysis, with D.Drasin and R. S. Laugesen, Cambridge University Press, Cambridge, 2019. · Zbl 1509.32001
[27] G. Bianchi, R. J. Gardner, and P. Gronchi, Symmetrization in geometry, Adv. Math. 306 (2017), 51-88. · Zbl 1366.52003
[28] G. Bianchi, R. J. Gardner, and P. Gronchi, Convergence of symmetrization processes, preprint, arXiv:1908.03259v2.
[29] G. Bianchi, R. J. Gardner, P. Gronchi, and M. Kiderlen, Rearrangement and polarization, Adv. Math. 374, 107380, 51 pp. · Zbl 1448.28001
[30] G. Bianchi, R. J. Gardner, P. Gronchi, and M. Kiderlen, Smoothing rearrangements and the Pólya-Szegő inequality, in preparation.
[31] A. Cianchi, E. Lutwak, D. Yang and G. Zhang, Affine Moser-Trudinger and Morrey-Sobolev inequalities, Calc. Var. 36 (2009), 419-436. · Zbl 1202.26029
[32] G. Talenti, The art of rearranging, Milan J. Math. 84 (2016), 105-157. References · Zbl 1364.35006
[33] G. Berck, Convexity of Lp-intersection bodies. Adv. Math. 222 (2009), 920-936. · Zbl 1179.52005
[34] H. Busemann, Volume in terms of concurrent cross-sections. Pacific J. Math. 3 (1953), 1-12. · Zbl 0050.16702
[35] S. Campi, P. Gronchi, The Lp-Busemann-Petty centroid inequality. Adv. Math. 167 (2002), 128-141. · Zbl 1002.52005
[36] D. Cordero-Erausquin, M. Fradelizi, G. Paouris, P. Pivovarov, Volume of the polar of random sets and shadow systems. Math. Ann. 362 (2015), 1305-1325. · Zbl 1366.52008
[37] A. Koldobsky, Fourier analysis in convex geometry. Mathematical Surveys and Monographs, 116. American Mathematical Society, Providence, RI, 2005. · Zbl 1082.52002
[38] C. Haberl, M. Ludwig, A characterization of Lp intersection bodies. Int. Math. Res. Not. 2006, Art. ID 10548, 29 pp. · Zbl 1115.52006
[39] E. Lutwak, Selected affine isoperimetric inequalities. Handbook of convex geometry, Vol. A, B, 151-176, North-Holland, Amsterdam, 1993. · Zbl 0847.52006
[40] E. Lutwak, D. Yang, G. Zhang, Lp affine isoperimetric inequalities. J. Differential Geom. 56 (2000), 111-132. · Zbl 1034.52009
[41] E. Lutwak, G. Zhang, Blaschke-Santaló inequalities. J. Differential Geom. 47 (1997), 1-16. · Zbl 0906.52003
[42] P. Mankiewicz, N. Tomczak-Jaegermann, Quotients of finite-dimensional Banach spaces; random phenomena. Handbook of the geometry of Banach spaces, Vol. 2, 1201-1246, North-Holland, Amsterdam, 2003. · Zbl 1057.46010
[43] P. Nayar, T. Tkocz, On a convexity property of sections of the cross-polytope. Proc. Amer. Math. Soc. 148 (2020), no. 3, 1271-1278. · Zbl 1447.52013
[44] G. Paouris, Concentration of mass on convex bodies. Geom. Funct. Anal. 16 (2006), 1021-1049. · Zbl 1114.52004
[45] G. Paouris, P. Pivovarov, A probabilistic take on isoperimetric-type inequalities. Adv. Math. 230 (2012), 1402-1422. · Zbl 1260.52006
[46] Petty, C. M. Centroid surfaces. Pacific J. Math. 11 (1961), 1535-1547. · Zbl 0103.15604
[47] V. Yaskin, M. Yaskina, Centroid bodies and comparison of volumes. Indiana Univ. Math. J. 55 (2006), no. 3, 1175-1194. · Zbl 1102.52005
[48] K. Ball, K. Böröczky, Stability of the Prékopa-Leindler inequality, Mathematika, 56 (2010), 339-356. · Zbl 1205.39024
[49] M. Barchiesi, V. Julin, Robustness of the Gaussian concentration inequality and the Brunn-Minkowski inequality, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 80, 12 pp. · Zbl 1378.60042
[50] C. Borell, Convex set functions in d-space, Period. Math. Hungar. 6 (1975), 111-136. · Zbl 0274.28009
[51] K. Böröczky, A. De, Stability of the Prékopa-Leindler inequality for log-concave functions, J. Differential Equations, 298 (2021), 298-322. · Zbl 1511.35204
[52] D. Bucur, I. Fragalà, Lower bounds for the Prékopa-Leindler deficit by some distances mod-ulo translations, J. Convex Anal., 21 (2014), 289-305. · Zbl 1326.26032
[53] Y. Chen, An Almost Constant Lower Bound of the Isoperimetric Coefficient in the KLS Conjecture, GAFA, 31 (2021), 34-61. · Zbl 1495.52003
[54] M. Christ, An approximate inverse Riesz-Sobolev inequality, preprint, available online at http://arxiv.org/abs/1112.3715, 2012.
[55] S. Dubuc, Critères de convexité et inégalités integralés Ann. Inst. Fourier Grenoble, 27 (1) (1977), 135-165. · Zbl 0331.26008
[56] A. Figalli, D. Jerison, Quantitative stability for the Brunn-Minkowski inequality, Adv. Math., 314 (2017), 1-47. · Zbl 1380.52010
[57] A. Figalli, F. Maggi, C. Mooney, The sharp quantitative Euclidean concentration inequality, Camb. J. Math., 6 (2018), no. 1, 59-87. · Zbl 1385.39005
[58] A. Figalli, F. Maggi, A. Pratelli: A refined Brunn-Minkowski inequality for convex sets, Annales de IHP, 26 (2009), 2511-2519. · Zbl 1192.52015
[59] A. Figalli, F. Maggi, A. Pratelli, A mass transportation approach to quantitative isoperi-metric inequalities, Invent. Math., 182 (2010), 167-211. · Zbl 1196.49033
[60] P. van Hintum, H. Spink, M. Tiba, Sharp Stability of Brunn-Minkowski for Homothetic Regions, J. Eur. Math. Soc. (JEMS), accepted. arXiv:1907.13011
[61] P. van Hintum, H. Spink, M. Tiba, Sharp quantitative stability of the planar Brunn-Minkowski inequality, J. Eur. Math. Soc. (JEMS), accepted. arXiv:1911.11945
[62] A.V. Kolesnikov, E. Milman, Local Lp-Brunn-Minkowski inequalities for p < 1, Memoirs AMS, accepted. arXiv:1711.01089
[63] L. Leindler, On a certain converse of Hölder’s inequality. II, Acta Sci. Math. (Szeged) 33 (1972), 217-223. · Zbl 0245.26011
[64] A. Prékopa, Logarithmic concave measures with application to stochastic programming, Acta Sci. Math. (Szeged) 32 (1971), 301-316. · Zbl 0235.90044
[65] A. Prékopa, On logarithmic concave measures and functions, Acta Sci. Math. (Szeged) 34 (1973), 335-343. · Zbl 0264.90038
[66] K. Berlow, M. C. Brandenburg, C. Meroni, I. Shankar, Intersection Bodies of Polytopes, arXiv:2110.05996, to appear in Beiträge zur Algebra und Geometrie.
[67] W. Blaschke, Vorlesungenüber Differentialgeometrie. II, Springer-Verlag, Berlin, (1923). · JFM 49.0499.01
[68] J. Bochnak, M. Coste, M. F. Roy. Real Algebraic Geometry, Springer, (2013).
[69] E. D. Bolker, The zonoid problem, Amer. Math. Monthly, 78(5) (1971), 529-531.
[70] D. A. Cox, J. Little, D. O’Shea, Ideals, Varieties, and Algorithms, Springer, (1991).
[71] F. Gesmundo, C. Meroni, The Geometry of Discotopes, arXiv:2111.01241, to appear in Le Matematiche.
[72] T. P. Le, C. Meroni, B. Sturmfels, R. F. Werner, T. Ziegler, Quantum Correlations in the Minimal Scenario, arXiv:2111.06270.
[73] L. Mathis, C. Meroni, Fiber Convex Bodies, arXiv:2105.12406.
[74] M. Micha lek, B. Sturmfels, Invitation to Nonlinear Algebra, volume 211 of Graduate Studies in Mathematics, AMS, (2021). · Zbl 1477.14001
[75] D. Plaumann, R. Sinn, J. L. Wesner, Families of faces and the normal cycle of a convex semi-algebraic set, arXiv:2104.13306.
[76] M. Ramana, A. J. Goldman, Some geometric results in semidefinite programming, Journal of Global Optimization, 7(1) (1995), 33-50. · Zbl 0839.90093
[77] K. Ranestad, B. Sturmfels, The convex hull of a variety, In Petter Brändén, Mikael Passare, and Mihai Putinar, editors, Notions of Positivity and the Geometry of Polynomials, Springer Verlag, Basel, (2011), 331-344. · Zbl 1253.14055
[78] K. Ranestad, B. Sturmfels, On the convex hull of a space curve, Advances in Geometry, 12(1) (2012), 157-178. · Zbl 1245.14033
[79] J. Saunderson, V. Chandrasekaran, Terracini convexity, arXiv:2010.00805.
[80] W. Weil, Blaschkes Problem der lokalen Charakterisierung von Zonoiden, Arch. Math., 29(1) (1977), 655-659. · Zbl 0382.52006
[81] W. Weil, Zonoide und verwandte Klassen konvexer Körper, Monatsh. Math., 94(1) (1982), 73-84. · Zbl 0483.52001
[82] S. Alesker, Valuations on convex functions and convex sets and Monge-Ampère operators, Adv. Geom. 19 (2019), 313-322. · Zbl 1445.52011
[83] S. Alesker, S. Artstein-Avidan, D. Faifman and V. Milman, A characterization of product preserving maps with applications to a characterization of the Fourier transform, Illinois J. Math. 54 (2010), 1115-1132 (2012). · Zbl 1272.42004
[84] S. Artstein-Avidan, H. König and V. Milman, The chain rule as a functional equation, J. Funct. Anal. 259 (2010), 2999-3024. · Zbl 1203.39014
[85] S. Artstein-Avidan and V. Milman, The concept of duality for measure projections of convex bodies, J. Funct. Anal. 254 (2008), 2648-2666. · Zbl 1145.26003
[86] S. Artstein-Avidan and V. D. Milman, The concept of duality in convex analysis, and the characterization of the Legendre transform, Ann. of Math. (2009), 661-674. · Zbl 1173.26008
[87] K. J. Böröczky and R. Schneider, A characterization of the duality mapping for convex bodies, Geom. Funct. Anal. 18 (2008), 657-667. · Zbl 1168.52002
[88] A. Colesanti, M. Ludwig and F. Mussnig, Minkowski valuations on convex functions, Calc. Var. Partial Differential Equations 56 (2017), Art. 162, 29 pp. · Zbl 1400.52014
[89] A. Colesanti, M. Ludwig and F. Mussnig, The Hadwiger theorem on convex functions. I, arXiv preprint arXiv:2009.03702 (2020).
[90] P. M. Gruber, The endomorphisms of the lattice of norms in finite dimensions, Abh. Math. Sem. Univ. Hamburg 62 (1992), 179-189. · Zbl 0779.52006
[91] C. Haberl and L. Parapatits, The centro-affine Hadwiger theorem, J. Amer. Math. Soc. 27 (2014), 685-705. · Zbl 1319.52006
[92] H. Hadwiger, Vorlesungenüber Inhalt, Oberfläche und Isoperimetrie, Springer, Berlin, 1957. · Zbl 0078.35703
[93] J. Li and D. Ma, Laplace transforms and valuations, J. Funct. Anal. 272 (2017), 738-758. · Zbl 1353.44001
[94] M. Ludwig, Minkowski valuations, Trans. Amer. Math. Soc. 357 (2005), 4191-4213. · Zbl 1077.52005
[95] M. Ludwig, Intersection bodies and valuations, Amer. J. Math. 128 (2006), 1409-1428. · Zbl 1115.52007
[96] M. Ludwig, Minkowski areas and valuations, J. Differential Geom. 86 (2010), 133-161. · Zbl 1215.52004
[97] A. Tsang, Valuations on L p -Spaces, Int. Math. Res. Not. 20 (2010), 3993-4023. · Zbl 1211.52013
[98] A. Tsang, Minkowski valuations on L p -spaces, Trans. Amer. Math. Soc. 364 (2012), 6159-6186. · Zbl 1279.52008
[99] W. Wang and L. Liu, Fourier transform and valuations, J. Math. Anal. Appl. 470 (2019), 1167-1184. · Zbl 1412.42013
[100] S. Alesker, Valuations on convex functions and convex sets and Monge-Ampère operators, Adv. Geom. 19 (2019), 313-322. · Zbl 1445.52011
[101] S. Bobkov, A. Colesanti, I. Fragalá, Quermassintegrals of quasi-concave functions and gen-eralized Prékopa-Leindler inequalities, Manuscripta Math. 143 (2014), 131-169. · Zbl 1290.26019
[102] A. Colesanti, N. Lombardi, Valuations on the space of quasi-concave functions, Geomet-ric Aspects of Functional Analysis, 71-105. Lecture Notes in Mathematics, 2169. Cham: Springer International Publishing, 2017. · Zbl 1375.52010
[103] A. Colesanti, N. Lombardi, L. Parapatits, Translation invariant valuations on quasi-concave functions, Stud. Math. 243 (2018), 79-99. · Zbl 1471.26005
[104] A. Colesanti, M. Ludwig, and F. Mussnig, A homogeneous decomposition theorem for val-uations on convex functions, J. Funct. Anal. 279 (2020), 108-573. · Zbl 1446.26010
[105] A. Colesanti, M. Ludwig, and F. Mussnig, The Hadwiger theorem on convex functions. I, arXiv:2009.03702 (2020).
[106] A. Colesanti, M. Ludwig, and F. Mussnig, The Hadwiger theorem on convex functions. II, arXiv:2109.09434 (2021).
[107] A. Colesanti, M. Ludwig, and F. Mussnig, The Hadwiger theorem on convex functions. III, arXiv:2111.05648 (2021).
[108] J. Knoerr, The support of dually epi-translation invariant valuations on convex functions, J. Funct. Anal. 281 (2021), 109-059. · Zbl 1487.52023
[109] V. Milman, L. Rotem, Mixed integrals and related inequalities, J. Funct. Anal., 264 (2013), 570-604. · Zbl 1267.26021
[110] Schneider, R., Convex Bodies: The Brunn-Minkowski Theory, vol. 151, Second Expanded Edition. Encyclopedia of Mathematics and its Applications, Cambridge: Cambridge Univer-sity Press, 2014. References · Zbl 1287.52001
[111] S. Alesker, Valuations on convex functions and convex sets and Monge-Ampère operators, Adv. Geom. 19 (2019), 313-322. · Zbl 1445.52011
[112] A. Colesanti, M. Ludwig, F. Mussnig, The Hadwiger theorem on convex functions. I, arXiv:2009.03702.
[113] A. Colesanti, M. Ludwig, F. Mussnig, The Hadwiger theorem on convex functions. III, arXiv:2111.05648.
[114] J. Knoerr, Smooth and mixed Hessian valuations on convex functions, arXiv:2006.12933. References
[115] Kannan, Lovász and Simonovits, Isoperimetric problems for convex bodies and a localization lemma, Discrete & Computational Geometry (1995), 541-559. · Zbl 0824.52012
[116] Eldan, Thin shell implies spectral gap up to polylog via a stochastic localization scheme, Geometric and Functional Analysis (2013), 532-569 · Zbl 1277.52013
[117] S. Artstein-Avidan and V. Milman, The concept of duality in convex analysis, and the characterization of the Legendre transform, Annals of mathematics, (2009), pp. 661-674. · Zbl 1173.26008
[118] S. Artstein-Avidan and V. Milman, Hidden structures in the class of convex functions and a new duality transform, Journal of the European Mathematical Society, vol. 13, no 4, (2011) pp. 975-1004. · Zbl 1221.26018
[119] J. Bertrand, A. Pratelli and M. Puel,Kantorovich potentials and continuity of total cost for relativistic cost functions, Journal de Mathématiques Pures et Appliquées, vol. 110, (2018), pp. 93-122. · Zbl 1381.49048
[120] V. Oliker, Embedding S n into R n+1 with given integral Gauss curvature and optimal mass transport on S n , Advances in Mathematics, vol. 213, no 2, (2007), pp. 600-620. · Zbl 1233.49024
[121] U. Bindini and L. De Pascale, Optimal transport with Coulomb cost and the semiclassical limit of density functional theory, Journal de l’École polytechnique-Mathématiques, vol. 4, (2017), pp. 909-934. · Zbl 1409.49010
[122] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Communications on pure and applied mathematics, vol. 44, no 4, (1991), pp. 375-417. · Zbl 0738.46011
[123] R. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math-ematical Journal, vol. 80, no 2, (1995), pp. 309-323. · Zbl 0873.28009
[124] R.T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific Journal of Mathematics, vol. 17, no 3, (1966), pp. 497-510. · Zbl 0145.15901
[125] J. Rochet, A necessary and sufficient condition for rationalizability in a quasi-linear context, Journal of mathematical Economics, vol. 16, no 2, (1987), pp. 191-200. · Zbl 0628.90003
[126] L. Rüschendorf, On c-optimal random variables, Statistics & probability letters, vol. 27, no 3, (1996), pp. 267-270. · Zbl 0847.62046
[127] S. Artstein-Avidan, S. Sadovsky and K. Wyczesany, A Rockafellar-type theorem for non-traditional costs, Advances in Mathematics, vol. 395, (2022). · Zbl 1480.49043
[128] S. Artstein-Avidan, S. Sadovsky and K. Wyczesany, Optimal measure transportation with respect to non-traditional costs, arXiv:2104.04838 (submitted), (2021).
[129] L. Ambrosio and A. Pratelli, Existence and stability results in the L 1 theory of optimal transportation, Optimal transportation and applications, (2003), pp. 123-160. · Zbl 1065.49026
[130] S. Artstein-Avidan, S. Sadovsky and K. Wyczesany, A zoo of set dualities, arXiv:2110.11308, (2021).
[131] Caffarelli, L. A., Monotonicity properties of optimal transportation and the FKG and related inequalities. Comm. Math. Phys., Vol. 214, no. 3, (2000), 547-563. · Zbl 0978.60107
[132] Cattiaux, P., Guillin, A., On the Poincaré constant of log-concave measures. Geometric aspects of functional analysis -Israel seminar. Lecture Notes in Math., Vol. 2256, Springer, (2020), 171-217. · Zbl 1453.60049
[133] Klartag, B., Putterman, E., Spectral monotonicity under Gaussian convolution. To appear in Annales de la Faculté des Sciences de Toulouse. Available under https://arxiv.org/abs/2107.09496” Kohler-Jobin meets Ehrhard Galyna Livshyts (joint work with Orli Herscovici)
[134] Consider a log-concave measure µ on R n with density e −V , for some convex func-tion V , and its associated Laplacian L• = ∆ • − ∇•, ∇V .
[135] L. Brasco, On torsional rigidity and principal frequencies: an invitation to the Kohler-Jobin rearrangement technique. ESAIM Control Optim. Calc. Var. 20 (2014), no. 2, 315-338. · Zbl 1290.35160
[136] A. Burchard, A short course on rearrangement inequalities, lecture notes, 2009.
[137] C. Borell, The Brunn-Minkowski inequality in Gauss spaces, Invent. Math 30 (1975), 207-216. · Zbl 0292.60004
[138] E.A. Carlen and C. Kerce, On the cases of equality in Bobkov’s inequality and Gaussian rearrangement. Calc. Var. 13 (2001), 1-18. · Zbl 1009.49029
[139] A. Ehrhard, Symetrisation dans l’espace de Gauss, Math. Scand. 53 (1983), 281-301. · Zbl 0542.60003
[140] A. Ehrhard, Elements extremaux pour les inegalites de Brunn-Minkowski Gaussiennes, Ann. Inst. H. Poincare Probab. Statist., 22(2):149-168, 1986. · Zbl 0595.60020
[141] G. Faber, Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gle-icher Spannung die kreisförmige den tiefsten Grundton gibt Sitzungsber. Bayer. Akad. Wiss. München, Math.-Phys. Kl. (1923) pp. 169-172. · JFM 49.0342.03
[142] B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Mathe-matics 1150, Springer, Berlin, 1985. · Zbl 0593.35002
[143] S. Kesavan, Symmetrization and Applications, Series in Analysis, Volume 3, World Scientific, (2006), ISBN 981-256-733-X · Zbl 1110.35002
[144] M.-T. Kohler-Jobin, Symmetrization with equal Dirichlet integrals. SIAM J. Math. Anal. 13 (1982), no. 1, 153-161. · Zbl 0484.35006
[145] M.-T. Kohler-Jobin, Une méthode de comparaison isopérimétrique de fonctionnelles de do-maines de la physique mathématique. I. Une démonstration de la conjecture isopérimétrique P λ 2 ≥ πj 4
[146] /2 de Pólya et Szegö. (French) Z. Angew. Math. Phys. 29 (1978), no. 5, 757-766. · Zbl 0427.73056
[147] E. Krahn,Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises Math. Ann., 94 (1925) pp. 97-100. · JFM 51.0356.05
[148] E. Krahn,Über Minimaleigenschaften der Kugel in drei und mehr Dimensionen Acta Comm. Univ. Tartu (Dorpat), A9 (1926) pp. 1-44 (English transl.:Ü. Lumiste and J. Pee-tre (eds.), Edgar Krahn, 1894-1961, A Centenary Volume, IOS Press, 1994, Chap. 6, pp. 139-174)
[149] E. H Lieb, M. Loss, Analysis, Second Edition, Graduate studies in mathematics, vol. 14, 2001, American Mathematical Society, Providence · Zbl 0966.26002
[150] G.V. Livshyts, On a conjectural symmetric version of Ehrhard’s inequality. arXiv:2103.11433.
[151] G. Pólya, G. Szegö, Isoperimetric Inequalities in Mathematical Physics. Annals of Mathe-matics Studies, No. 27 Princeton University Press, Princeton, N. J., 1951. xvi+279 pp. · Zbl 0044.38301
[152] V. N. Sudakov and B. S. Tsirel’son, Extremal properties of half-spaces for spherically invari-ant measures. Problems in the theory of probability distributions, II. Zap. Nauch. Leningrad Otdel. Mat. Inst. Steklov 41 (1974), 14-24 (in Russian). · Zbl 0351.28015
[153] J.L. Vazquez, The Porous Medium Equation, Mathematical Theory, Oxford University Press (2007). · Zbl 1107.35003
[154] S. Alesker, A. Bernig, and F.E. Schuster, Harmonic analysis of translation invariant valu-ations, Geom. Funct. Anal. 21 (2011), 751-773. · Zbl 1228.53088
[155] S. Alesker, Kotrbatý’s theorem on valuations and geometric inequalities for convex bodies, Israel J. Math. (2021).
[156] J. Kotrbatý, On Hodge-Riemann relations for translation-invariant valuations Adv. Math. (2021). · Zbl 1475.52011
[157] J. Kotrbatý and T. Wannerer, On mixed Hodge-Riemann relations for translation-invariant valuations and Aleksandrov-Fenchel inequalities, Commun. Contemp. Math. (2021).
[158] Alesker, Semyon; Hard Lefschetz theorem for valuations, complex integral geometry, and unitarily invariant valuations. J. Differential Geom. 63 (2003), no. 1, 63-95. · Zbl 1073.52004
[159] Alesker, Semyon; Kotrbatý’s theorem on valuations and geometric inequalities for convex bodies. Israel J. Math., to appear. · Zbl 1497.52011
[160] Bernig, Andreas; Fu, Joseph H. G.; Convolution of convex valuations. Geom. Dedicata 123 (2006), 153-169. · Zbl 1117.53054
[161] Bernig, Andreas; Bröcker, Ludwig; Valuations on manifolds and Rumin cohomology. J. Dif-ferential Geom. 75 (2007), no. 3, 433-457. · Zbl 1117.58005
[162] Kotrbatý, Jan On Hodge-Riemann relations for translation-invariant valuations. Adv. Math. 390 (2021), Paper No. 107914, 28 pp. · Zbl 1475.52011
[163] Kotrbatý, Jan; Wannerer, Thomas; On mixed Hodge-Riemann relations for translation-invariant valuations and Aleksandrov-Fenchel inequalities. Commun. Contemp. Math., to appear. · Zbl 1496.52011
[164] Andreas Bernig and Joseph H. G. Fu. Hermitian integral geometry. Ann. of Math., 173:907-945, 2011. · Zbl 1230.52014
[165] Andreas Bernig, Joseph H. G. Fu, and Gil Solanes. Integral geometry of complex space forms. Geom. Funct. Anal., 24(2):403-492, 2014. · Zbl 1298.53074
[166] Joseph H. G. Fu. Structure of the unitary valuation algebra. J. Differential Geom., 72(3):509-533, 2006. · Zbl 1096.52003
[167] Christer Borell. Convex set functions in d-space. Periodica Mathematica Hungarica, 6(2):111-136, 1975. · Zbl 0307.28009
[168] Károly J. Böröczky, Erwin Lutwak, Deane Yang, and Gaoyong Zhang. The log-Brunn-Minkowski inequality. Advances in Mathematics, 231(3-4):1974-1997, oct 2012. · Zbl 1258.52005
[169] Dario Cordero-Erausquin, Matthieu Fradelizi, and Bernard Maurey. The (B) conjecture for the Gaussian measure of dilates of symmetric convex sets and related problems. Journal of Functional Analysis, 214(2):410-427, sep 2004. · Zbl 1073.60042
[170] Dario Cordero-Erausquin and Liran Rotem. Improved log-concavity for rotationally invari-ant measures of symmetric convex sets. arxiv:2111.05110, nov 2021.
[171] Alexandros Eskenazis and Georgios Moschidis. The dimensional Brunn-Minkowski inequal-ity in Gauss space. Journal of Functional Analysis, 280(6):108914, mar 2021. · Zbl 1456.52011
[172] Alexandros Eskenazis, Piotr Nayar, and Tomasz Tkocz. Gaussian mixtures: Entropy and geometric inequalities. The Annals of Probability, 46(5):2908-2945, sep 2018. · Zbl 1428.60036
[173] Richard J. Gardner and Artem Zvavitch. Gaussian Brunn-Minkowski inequalities. Transac-tions of the American Mathematical Society, 362(10):5333-5333, oct 2010. · Zbl 1205.52002
[174] Alexander Kolesnikov and Galyna Livshyts. On the Gardner-Zvavitch conjecture: Symmetry in inequalities of Brunn-Minkowski type. Advances in Mathematics, 384:107689, jun 2021. · Zbl 1479.52014
[175] Alexander Kolesnikov and Emanuel Milman. Local L p -Brunn-Minkowski inequalities for p < 1. arxiv:1711.01089, nov 2017.
[176] Alexander Kolesnikov and Emanuel Milman. Poincaré and Brunn-Minkowski inequalities on the boundary of weighted Riemannian manifolds. American Journal of Mathematics, 140(5):1147-1185, 2018. · Zbl 1408.53047
[177] Rafa l Lata la. On some inequalities for Gaussian measures. In Proceedings of the Interna-tional Congress of Mathematicians, Beijing, volume II, pages 813-822. Higher Ed. Press, Beijing, 2002. · Zbl 1015.60011
[178] Galyna Livshyts. A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures. arxiv:2107.00095, jun 2021.
[179] Galyna Livshyts, Arnaud Marsiglietti, Piotr Nayar, and Artem Zvavitch. On the Brunn-Minkowski inequality for general measures with applications to new isoperimetric-type inequalities. Transactions of the American Mathematical Society, 369(12):8725-8742, apr 2017. · Zbl 1376.52014
[180] Christos Saroglou. More on logarithmic sums of convex bodies. Mathematika, 62(03):818-841, may 2016. · Zbl 1352.52001
[181] G. Bonnet, A. Gusakova, C. Thäle, D. Zaporozhets, Sharp inequalities for the mean distance of random points in convex bodies, Adv. Math. 386 (2021), article 107813, 27 pages. References · Zbl 1467.52010
[182] F. M. Baêta, J. Haddad, On explicit representations of isotropic measures in John and Löwner positions, arXiv:2111.03624
[183] S. Artstein-Avidan, D. Katzin Isotropic measures and maximizing ellipsoids between John and Loewner, Proceedings of the American Mathematical Society, 146(12):5379-5390, 2018 · Zbl 1401.52017
[184] F. John. Extremum problems with inequalities as subsidiary conditions, studies and essays presented to R. Courant on his 60th birthday, january 8, 1948, 1948. References
[185] B. He, G. Leng, K. Li, Projection problems for symmetric polytopes, Adv. Math. 207 (2006) 73-90. · Zbl 1111.52012
[186] M. Henk, E. Linke, Cone-volume measures of polytopes, Adv. Math. 253 (2014) 50-62. · Zbl 1308.52007
[187] Yude Liu, Q. Sun, G. Xiong, Sharp affine isoperimetric inequalities for the volume decom-position functionals of polytopes, Adv. Math. 389 (2021), Paper No. 107902, 29 pp. · Zbl 1479.52015
[188] E. Lutwak, D. Yang, G. Zhang, A new affine invariant for polytopes and Schneider’s pro-jection problem, Trans. Amer. Math. Soc. 353 (2001) 1767-1779. · Zbl 0971.52011
[189] G. Xiong, Extremum problems for the cone volume functional of convex polytopes, Adv. Math. 225 (2010) 3214-3228. · Zbl 1213.52011
[190] A. V. Kolesnikov and E. M. Werner, Blaschke-Santaló inequality for many functions and geodesic barycenters of measures, Advances in Math., to appear. · Zbl 1482.52004
[191] A.D, Aleksandrov, Existence and uniqueness of a convex surface with given integral curva-ture, Acad. Sci. USSR (N.S.) 35 (1942), 131-134. · Zbl 0061.37604
[192] H. Federer, Curvature Measures, Trans. Amer. Math 93 (1959), 418-491. · Zbl 0089.38402
[193] Y. Huang, E. Lutwak, D. Yang, G. Zhang, Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems, Acta Math. 216 (2016), 325-388. · Zbl 1372.52007
[194] Y. Huang, E. Lutwak, D. Yang, G. Zhang, The Lp-Aleksandrov problem for Lp-integral curvature , J. Diff. Geom. 110 (2018), 1-29. · Zbl 1404.35139
[195] V. Oliker, Embedding S n−1 into R n+1 with given integral Gauss curvature and optimal mass transport on S n−1 , Adv. Math. 213 (2007), 600-620. · Zbl 1233.49024
[196] Y. Zhao, The Lp-Aleksandrov problem for origin-symmetric polytopes, Proc. Amer. Math. Soc. 147 (2019), 4477-4492. · Zbl 1423.52012
[197] I. Bárány and Z. Füredi Computing the volume is difficult, Discrete and Computational Geometry 2 (1987), no. 4, 319-326. · Zbl 0628.68041
[198] M. Grötschel, L. Lovász and A. Schrijver, Geometric Algorithms and Combinatorial Op-timization, Algorithms and Combinatorics: Study and Research Texts, 2, Springer-Verlag, Berlin, 1988. · Zbl 0634.05001
[199] K. J. Böröczky, E. Lutwak, D. Yang, and G. Zhang. The log-Brunn-Minkowski inequality. Adv. Math., 231(3-4):1974-1997, 2012. · Zbl 1258.52005
[200] K. J. Böröczky, E. Lutwak, D. Yang, and G. Zhang. The logarithmic Minkowski problem. J. Amer. Math. Soc., 26(3):831-852, 2013. · Zbl 1272.52012
[201] S. Chen, Y. Huang, Q.-r. Li, and J. Liu. The Lp-Brunn-Minkowski inequality for p < 1. Adv. Math., 368:107166, 2020. · Zbl 1440.52013
[202] A. Colesanti. From the Brunn-Minkowski inequality to a class of Poincaré-type inequalities. Commun. Contemp. Math., 10(5): 765-772, 2008. · Zbl 1157.52002
[203] A. V. Kolesnikov and E. Milman. Poincaré and Brunn-Minkowski inequalities on weighted Riemannian manifolds with boundary. arxiv.org/abs/1310.2526v1, 2013.
[204] A. V. Kolesnikov and E. Milman. Brascamp-Lieb-type inequalities on weighted Riemannian manifolds with boundary. J. Geom. Anal., 27(2):1680-1702, 2017. · Zbl 1372.53040
[205] A. V. Kolesnikov and E. Milman. Poincaré and Brunn-Minkowski inequalities on the bound-ary of weighted Riemannian manifolds. Amer. J. Math., 140(5):1147-1185, 2018. · Zbl 1408.53047
[206] A. V. Kolesnikov and E. Milman. Local L p -Brunn-Minkowski inequalities for p < 1. arxiv.org/abs/1711.01089, to appear in Mem. Amer. Math. Soc., 2017.
[207] E. Milman. A sharp centro-affine isospectral inequality of Szegö-Weinberger type and the L p -Minkowski problem. arxiv.org/abs/2103.02994, 2021.
[208] E. Milman. Centro-Affine Differential Geometry and the log-Minkowski problem. arxiv.org/abs/2104.12408, 2021.
[209] E. Putterman. Equivalence of the local and global versions of the L p -Brunn-Minkowski inequality. J. Func. Anal., 280(9):1089562019, 2021. References · Zbl 1461.52010
[210] K. Ball, Cube slicing in R n . Proc. Amer. Math. Soc. 97 (1986), no. 3, 465-473. · Zbl 0601.52005
[211] K. Gurushankar, G. Chasapis, T. Tkocz, Sharp bounds on p-norms for sums of independent uniform random variables, 0 < p < 1, to appear in J. Anal. Math. (2021), arXiv:2105.14079.
[212] G. Chasapis, H. König, T. Tkocz, From Ball’s cube slicing inequality to Khinchin-type inequalities for negative moments, J. Funct. Anal. 281 (2021), no. 9, Paper No. 109185, 23 pp. · Zbl 1482.52012
[213] G. Chasapis, P. Nayar, T. Tkocz, Slicing ℓp-balls reloaded: stability, planar sections in ℓ 1 , preprint (2021), arXiv:2109.05645.
[214] H. Hadwiger, Gitterperiodische Punktmengen und Isoperimetrie. Monatsh. Math. 76 (1972), 410-418. · Zbl 0248.52012
[215] U. Haagerup, The best constants in the Khintchine inequality. Studia Math. 70 (1981), no. 3, 231-283. · Zbl 0501.46015
[216] D. Hensley, Slicing the cube in R n and probability (bounds for the measure of a central cube slice in R n by probability methods). Proc. Amer. Math. Soc. 73 (1979), no. 1, 95-100. · Zbl 0368.52007
[217] A. Khintchine,Über dyadische Brüche. Math. Z. 18 (1923), no. 1, 109-116. · JFM 49.0132.01
[218] R. Lata la, K. Oleszkiewicz, A note on sums of independent uniformly distributed random variables. Colloq. Math. 68 (1995), no. 2, 197-206. · Zbl 0821.60027
[219] J. Melbourne, C. Roberto, Quantitative form of Ball’s Cube slicing in R n and equality cases in the min-entropy power inequality, preprint (2021), arXiv:2109.03946.
[220] F. L. Nazarov, A. N. Podkorytov, Ball, Haagerup, and distribution functions, Complex analysis, operators, and related topics, 247-267, Oper. Theory Adv. Appl., 113 (2000), Birkhäuser, Basel. · Zbl 0934.00031
[221] E. Milman, A. Yehudayoff, Sharp Isoperimetric Inequalities for Affine Quermassintegrals, https://arxiv.org/abs/2005.04769
[222] S. Brazitikos, F. MacIntyre, Vector-valued Maclaurin inequalities , to appear in Communi-cations in Contemporary Mathematics, https://doi.org/10.1142/S0219199721500449. Face numbers of high-dimensional Poisson polytopes Zakhar Kabluchko References · doi:10.1142/S0219199721500449.Facenumbersofhigh-dimensionalPoissonpolytopesZakharKabluchkoReferences
[223] Z. Kabluchko, Expected f -vector of the Poisson zero polytope and random convex hulls in the half-sphere, Mathematika, 66(4), 2020, pp. 1028-1053. · Zbl 1530.52008
[224] Z. Kabluchko, Face numbers of high-dimensional Poisson zero cells, arXiv: 2110.08201. References
[225] H. Auerbach, Sur un probléme de M. Ulam concernant l’équilibre des corps flottants, Studia Math. 7 (1938), no. 1, 121-142. · Zbl 0018.17504
[226] J. Bracho, L. Montejano and D. Oliveros, Carousels, Zindler curves and the float-ing body problem, Period. Math. Hungar. 49 (2004), no. 2, 9-23. · Zbl 1075.52003
[227] H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved problems in geometry. Prob-lem Books in Mathematics. Unsolved Problems in Intuitive Mathematics, II. Springer-Verlag, New York, 1991. · Zbl 0748.52001
[228] Ch.-J. de La Vallée Poussin, Lecons De Mécanique Analytique, Vol II. Paris, Gauthier-Villarséditeur, 55, Quai des Grands Augustins, Copyright by A. Uystpruyst, Louvain, 1925 (in French).
[229] K. J. Falconer, Applications of a Result on Spherical Integration to the Theory of Convex Sets, Amer. Math. Monthly 90 (1983), no. 10, 690-693. · Zbl 0529.52001
[230] D. I. Florentin, C. Schütt, E. M. Werner and N. Zhang, Convex floating bodies of Equilibrium, arXiv:2010.09006.
[231] R. J. Gardner, Geometric tomography. Second edition. Encyclopedia of Mathematics and its Applications, 58. Cambridge University Press, New York, 2006. · Zbl 1102.52002
[232] E. N. Gilbert, How things float, Amer. Math. Monthly 98 (1991), no. 3, 201-216. · Zbl 0735.76012
[233] S. Helgason, The Radon transform. Second edition. Progress in Mathematics, 5, Birkhäuser Boston Inc., Boston, MA, 1999. · Zbl 0932.43011
[234] H. Huang, B. Slomka and E. M. Werner, Ulam floating bodies, J. Lond. Math. Soc. (2) 100 (2019), no. 2, 425-446. · Zbl 1431.52006
[235] R. D. Mauldin, The Scottish book. Mathematics from the Scottish Café, with Selected Problems from The New Scottish book, Second Edition, Birkhäuser, 2015, ISBN 978-3-319-22896-9. · Zbl 1331.01039
[236] F. Nazarov, D. Ryabogin and A. Zvavitch, An asymmetric convex body with maximal sections of constant volume, J. Amer. Math. Soc. 27 (2014), no. 1, 43-68. · Zbl 1283.52005
[237] S. P. Olovjanischnikoff, Ueber eine kennzeichnende Eigenschaft des Ellipsoides, Uchenye Zapiski Leningrad State Univ., Math. Ser. 83 (1941), no. 12, 114-128. · Zbl 0061.37709
[238] D. Ryabogin, On bodies floating in equilibrium in every orientation, arXiv:2010.09565.
[239] R. Schneider, Functional equations connected with rotations and their geometric ap-plications. Enseign. Math. (2) 16 (1970), 297-305. · Zbl 0209.26502
[240] R. Schneider, Convex Bodies: the Brunn-Minkowski theory. Second expanded edition. Encyclopedia of Mathematics and its Applications, 151. Cambridge University Press, Cambridge, 2014. · Zbl 1287.52001
[241] S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York-London, 1960. · Zbl 0086.24101
[242] P. L. Várkonyi, Neutrally floating objects of density 1 2 in three dimensions, Stud.
[243] Appl. Math. 130 (2013), no. 3, 295-315. · Zbl 1316.76017
[244] P. L. Várkonyi, Floating body problems in two dimensions, Stud. Appl. Math. 122 (2009), no. 2, 195-218. · Zbl 1168.76011
[245] F. Wegner, Floating bodies of equilibrium, Stud. Appl. Math. 111 (2003), no. 2, 167-183. · Zbl 1141.76345
[246] F. Wegner, Floating bodies in equilibrium in 2D, the tire track problem and electrons in a parabolic magnetic fields, arXiv:physics/0701241v3 (2007).
[247] V. A. Zalgaller, Theory of envelopes, Izdat. “Nauka”, Moscow, 1975 (in Russian).
[248] N. E. Zhukovsky, Classical mechanics, Moscow, 1936 (in Russian).
[249] K. Zindler,Über konvexe Gebilde II, Monatsh. Math. Phys. 31 (1921), 25-57. Reporter: Katarzyna Wyczesany
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.