On subspace concentration for dual curvature measures. (English) Zbl 07737683
Summary: We study subspace concentration of dual curvature measures of convex bodies \(K\) satisfying \(\gamma(- K) \subseteq K\) for some \(\gamma \in(0, 1]\). We present upper bounds on the subspace concentration depending on \(\gamma\), which, in particular, retrieves the known results in the symmetric setting. The proof is based on a unified approach to prove necessary subspace concentration conditions via the divergence theorem.
MSC:
| 52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |
References:
| [1] | Aleksandrov, A., Existence and uniqueness of a convex surface with a given integral curvature, C. R. (Dokl.) Acad. Sci. URSS, 35, 131-134 (1942) · Zbl 0061.37604 |
| [2] | Bianchi, G.; Böröczky, K. J.; Colesanti, A., Smoothness in the \(L_p\) Minkowski problem for \(p < 1\), J. Geom. Anal., 30, 1, 680-705 (2020) · Zbl 1445.52005 |
| [3] | Böröczky, K. J., The logarithmic Minkowski conjecture and the \(L_p\)-Minkowski problem, Harmonic Analysis and Convexity, 9 (2023), 83 pages · Zbl 1525.35156 |
| [4] | Böröczky, K. J.; De, A., Stable solution of the logarithmic Minkowski problem in the case of hyperplane symmetries, J. Differ. Equ., 298, 298-322 (2021) · Zbl 1511.35204 |
| [5] | Böröczky, K. J.; Hegedűs, P., The cone volume measure of antipodal points, Acta Math. Hung., 146, 2, 449-465 (2015) · Zbl 1363.52007 |
| [6] | Böröczky, K. J.; Hegedűs, P.; Zhu, G., On the discrete logarithmic Minkowski problem, Int. Math. Res. Not., 6, 1807-1838 (2016) · Zbl 1345.52002 |
| [7] | Böröczky, K. J.; Henk, M., Cone-volume measure of general centered convex bodies, Adv. Math., 286, 703-721 (2016) · Zbl 1334.52003 |
| [8] | Böröczky, K. J.; Henk, M., Cone-volume measure and stability, Adv. Math., 306, 24-50 (2017) · Zbl 1366.52007 |
| [9] | Böröczky, K. J.; Henk, M.; Pollehn, H., Subspace concentration of dual curvature measures of symmetric convex bodies, J. Differ. Geom., 109, 3, 411-429 (2018) · Zbl 1397.52005 |
| [10] | Böröczky, K. J.; Lutwak, E.; Yang, D.; Zhang, G., The logarithmic Minkowski problem, J. Am. Math. Soc., 26, 3, 831-852 (2013) · Zbl 1272.52012 |
| [11] | Böröczky, K. J.; Lutwak, E.; Yang, D.; Zhang, G.; Zhao, Y., The dual Minkowski problem for symmetric convex bodies, Adv. Math., 356, Article 106805 pp. (2019) · Zbl 1427.52006 |
| [12] | Chen, S.; Li, Q.-R., On the planar dual Minkowski problem, Adv. Math., 333, 87-117 (2018) · Zbl 1397.52002 |
| [13] | Chen, S.; Li, Q.-R.; Zhu, G., The logarithmic Minkowski problem for non-symmetric measures, Trans. Am. Math. Soc., 371, 4, 2623-2641 (2019) · Zbl 1406.52018 |
| [14] | Chou, K.-S.; Wang, X.-J., The \(L_p\)-Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math., 205, 1, 33-83 (2006) · Zbl 1245.52001 |
| [15] | Federer, H., Geometric Measure Theory (2014), Springer |
| [16] | Firey, W. J., p-means of convex bodies, Math. Scand., 10, 17-24 (1962) · Zbl 0188.27303 |
| [17] | Gardner, R. J., Geometric Tomography, vol. 6 (1995), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1042.52501 |
| [18] | Gardner, R. J.; Hug, D.; Weil, W.; Xing, S.; Ye, D., General volumes in the Orlicz-Brunn-Minkowski theory and a related Minkowski problem I, Calc. Var. Partial Differ. Equ., 58, 1, 1-35 (2019) · Zbl 1404.52004 |
| [19] | Gardner, R. J.; Hug, D.; Xing, S.; Ye, D., General volumes in the Orlicz-Brunn-Minkowski theory and a related Minkowski problem II, Calc. Var. Partial Differ. Equ., 59, 1, 1-33 (2020) · Zbl 1430.52001 |
| [20] | Groemer, H., On the symmetric difference metric for convex bodies, Beitr. Algebra Geom., 41, 1, 107-114 (2000) · Zbl 0951.52004 |
| [21] | Gruber, P. M., Convex and Discrete Geometry, vol. 336 (2007), Springer · Zbl 1139.52001 |
| [22] | Guo, L.; Xi, D.; Zhao, Y., The \(L_p\) Chord Minkowski problem for \(0 \leq p < 1 (2023)\), arXiv preprint |
| [23] | Hammer, P. C., The centroid of a convex body, Proc. Am. Math. Soc., 2, 4, 522-525 (1951) · Zbl 0043.16301 |
| [24] | Henk, M.; Linke, E., Cone-volume measures of polytopes, Adv. Math., 253, 50-62 (2014) · Zbl 1308.52007 |
| [25] | Henk, M.; Pollehn, H., Necessary subspace concentration conditions for the even dual Minkowski problem, Adv. Math., 323, 114-141 (2018) · Zbl 1383.52008 |
| [26] | Huang, Y.; Lutwak, E.; Yang, D.; Zhang, G., Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems, Acta Math., 216, 2, 325-388 (2016) · Zbl 1372.52007 |
| [27] | Huang, Y.; Lutwak, E.; Yang, D.; Zhang, G., The \(L_p\)-Aleksandrov problem for \(L_p\)-integral curvature, J. Differ. Geom., 110, 1, 1-29 (2018) · Zbl 1404.35139 |
| [28] | Hug, D.; Lutwak, E.; Yang, D.; Zhang, G., On the \(L_p\) Minkowski problem for polytopes, Discrete Comput. Geom., 33, 4, 699-715 (2005) · Zbl 1078.52008 |
| [29] | Jiang, Y.; Wu, Y., On the 2-dimensional dual Minkowski problem, J. Differ. Equ., 263, 6, 3230-3243 (2017) · Zbl 1387.52015 |
| [30] | Klee, V., Polyhedral sections of convex bodies, Acta Math., 103, 3-4, 243-267 (1960) · Zbl 0148.16203 |
| [31] | Kolesnikov, A. V., Mass transportation functionals on the sphere with applications to the logarithmic Minkowski problem, Mosc. Math. J., 20, 1, 67-91 (2020) · Zbl 1460.49037 |
| [32] | Li, Q.-R.; Sheng, W.; Wang, X.-J., Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems, J. Eur. Math. Soc., 22, 3, 893-923 (2020) · Zbl 1445.53066 |
| [33] | Lutwak, E., Dual mixed volumes, Pac. J. Math., 58, 2, 531-538 (1975) · Zbl 0273.52007 |
| [34] | Lutwak, E., The Brunn-Minkowski-Firey theory I: mixed volumes and the Minkowski problem, J. Differ. Geom., 38, 1, 131-150 (1993) · Zbl 0788.52007 |
| [35] | Lutwak, E., Selected Affine Isoperimetric Inequalities, Handbook of Convex Geometry, 151-176 (1993) · Zbl 0847.52006 |
| [36] | Lutwak, E., The Brunn-Minkowski-Firey theory ii: affine and geominimal surface areas, Adv. Math., 118, 2, 244-294 (1996) · Zbl 0853.52005 |
| [37] | Lutwak, E.; Yang, D.; Zhang, G., \( L_p\) dual curvature measures, Adv. Math., 329, 85-132 (2018) · Zbl 1388.52003 |
| [38] | Mui, S., On the \(L^p\) Aleksandrov problem for negative p, Adv. Math., 408(part A), Article 108573 pp. (2022), 26 · Zbl 1497.52005 |
| [39] | Pfeffer, W. F., The Divergence Theorem and Sets of Finite Perimeter (2012), CRC Press · Zbl 1258.28002 |
| [40] | Rockafellar, R. T., Convex Analysis, vol. 18 (1970), Princeton University Press · Zbl 0193.18401 |
| [41] | Schneider, R., Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and Its Applications (2013), Cambridge University Press |
| [42] | Stancu, A., The discrete planar \(L_0\)-Minkowski problem, Adv. Math., 167, 1, 160-174 (2002) · Zbl 1005.52002 |
| [43] | Stancu, A., On the number of solutions to the discrete two-dimensional \(L_0\)-Minkowski problem, Adv. Math., 180, 1, 290-323 (2003) · Zbl 1054.52001 |
| [44] | Süss, W., Über eine Affininvariante von Eibereichen, Arch. Math., 1, 2, 127-128 (1948) · Zbl 0031.27803 |
| [45] | Xiong, G., Extremum problems for the cone volume functional of convex polytopes, Adv. Math., 225, 6, 3214-3228 (2010) · Zbl 1213.52011 |
| [46] | Zhao, Y., The dual Minkowski problem for negative indices, Calc. Var. Partial Differ. Equ., 56, 2, 18 (2017) · Zbl 1392.52005 |
| [47] | Zhao, Y., Existence of solutions to the even dual Minkowski problem, J. Differ. Geom., 110, 3, 543-572 (2018) · Zbl 1406.52017 |
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