On the stability of the \(p\)-affine isoperimetric inequality. (English) Zbl 1433.52009
Summary: Employing the affine normal flow, we prove a stability version of the \(p\)-affine isoperimetric inequality for \(p\geq 1\) in \(\mathbb R^2\) in the class of origin-symmetric convex bodies. That is, if \(K\) is an origin-symmetric convex body in \(\mathbb R^2\) such that it has area \(\pi\) and its \(p\)-affine perimeter is close enough to the one of an ellipse with the same area, then, after applying a special linear transformation, \(K\) is close to an ellipse in the Hausdorff distance.
MSC:
| 52A40 | Inequalities and extremum problems involving convexity in convex geometry |
| 53E99 | Geometric evolution equations |
| 52A38 | Length, area, volume and convex sets (aspects of convex geometry) |
| 53A04 | Curves in Euclidean and related spaces |
| 53A15 | Affine differential geometry |
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
Keywords:
affine support function; affine normal flow; Hausdorff distance; stability of geometric inequalities; \(p\)-affine surface area; \(p\)-affine isoperimetric inequalityReferences:
| [1] | Andrews, B.: Contraction of convex hypersurfaces by their affine normal. J. Differ. Geom. 43, 207-230 (1996) · Zbl 0858.53005 |
| [2] | Andrews, B.: Motion of hypersurfaces by Gauss curvature. Pacific J. Math. 195(1), 1-34 (2000) · Zbl 1028.53072 · doi:10.2140/pjm.2000.195.1 |
| [3] | Andrews, B.: Singularities in crystalline curvature flows. Asian J. Math. 6, 101-121 (2002) · Zbl 1025.53038 |
| [4] | Ball, K., Böröczky, K.J.: Stability of the Prékopa-Leindler inequality. Mathematika 56, 339-356 (2010) · Zbl 1205.39024 · doi:10.1112/S002557931000063X |
| [5] | Ball, K., Böröczky, K.J.: Stability of some versions of the Prékopa-Leindler inequality. Monatshefte Math. 163, 1-14 (2011) · Zbl 1217.60014 · doi:10.1007/s00605-010-0222-z |
| [6] | Barthe, F., Böröczky, K.J., Fradelizi, M.: Stability of the functional forms of the Blaschke-Santaló inequality. arXiv:1206.0369v1 [math.MG] · Zbl 1288.52003 |
| [7] | Blaschke, W.: Über affine Geometrie I. Isoperimetrische Eigenschaften von Ellipse und Ellipsoid. Leipz. Ber. 68, 217-239 (1916) · JFM 46.1111.02 |
| [8] | Böröczky, K.J., Hug, D.: Stability of the reverse Blaschke-Santaló inequality for zonoids and applications. Adv. Appl. Math. 44, 309-328 (2010) · Zbl 1191.52002 · doi:10.1016/j.aam.2009.09.002 |
| [9] | Böröczky, K.J.: The stability of the Rogers-Shephard inequality. Adv. Math. 190, 47-76 (2005) · Zbl 1063.52002 · doi:10.1016/j.aim.2003.11.015 |
| [10] | Böröczky, K.J.: Stability of Blaschke-Santaló inequality and the affine isoperimetric inequality. Adv. Math. 225, 1914-1928 (2010) · Zbl 1216.52007 · doi:10.1016/j.aim.2010.04.014 |
| [11] | Böröczky, K.J., Makai, E. Jr.: On the volume product of planar polar convex bodies-upper estimates: the polygonal case and stability. In preparation · Zbl 1132.52300 |
| [12] | Cianchi, A., Lutwak, E., Yang, D., Zhang, G.: Affine Moser-Trudinger and Morrey-Sobolev inequalities. Calc. Var. PDEs. 36, 419-436 (2009) · Zbl 1202.26029 · doi:10.1007/s00526-009-0235-4 |
| [13] | Diskant, V.I.: Stability of the solution of a Minkowski equation. Sib. Mat. Zh. 14, 669-673 (1973) (in Russian). Engl. transl.: Siberian Math. J. 14, 466-473 (1974) · Zbl 0264.52007 |
| [14] | Hug, D.: Contributions to affine surface area. Manuscr. Math. 91, 283-301 (1996) · Zbl 0871.52004 · doi:10.1007/BF02567955 |
| [15] | Hug, D., Schneider, R.: A stability result for volume ratio. Isr. J. Math. 161, 209-219 (2007) · Zbl 1139.52012 · doi:10.1007/s11856-007-0079-6 |
| [16] | Groemer, H., Stability of geometric inequalities, 125-150 (1993), Amsterdam · Zbl 0789.52001 · doi:10.1016/B978-0-444-89596-7.50009-2 |
| [17] | Gruber, P.M.: Asymptotic estimates for best and stepwise approximation of convex bodies II. Forum Math. 5, 521-538 (1993) · Zbl 0788.41020 |
| [18] | Gruber, P.M.: Convex and Discrete Geometry. Grundlehren der Mathematischen Wissenschaften, vol. 336. Springer, Berlin (2007) · Zbl 1139.52001 |
| [19] | Ivaki, M.N.: Centro-affine curvature flows on centrally symmetric convex curves. Trans. Am. Math. Soc., to appear. arXiv:1205.6456v2 [math.DG] · Zbl 1298.53062 |
| [20] | Ivaki, M.N.: A flow approach to the L−2 Minkowski problem. Adv. Appl. Math. 50, 445-464 (2013) · Zbl 1261.53065 · doi:10.1016/j.aam.2012.09.003 |
| [21] | John, F., Extremum problems with inequalities as subsidiary conditions, Jan. 8, 1948, New York · Zbl 0034.10503 |
| [22] | Ludwig, M., Reitzner, M.: A classification of SL(n) invariant valuations. Ann. Math. 172, 1223-1271 (2010) · Zbl 1223.52007 · doi:10.4007/annals.2010.172.1223 |
| [23] | Ludwig, M.: General affine surface area. Adv. Math. 224, 2346-2360 (2010) · Zbl 1198.52004 · doi:10.1016/j.aim.2010.02.004 |
| [24] | Lutwak, E., Oliker, V.: On the regularity of solutions to a generalization of the Minkowski problem. J. Differ. Geom. 41, 227-246 (1995) · Zbl 0867.52003 |
| [25] | Lutwak, E.: The Brunn-Minkowski-Firey theory. I: Mixed volumes and the Minkowski problem. J. Differ. Geom. 38, 131-150 (1993) · Zbl 0788.52007 |
| [26] | Lutwak, E.: The Brunn-Minkowski-Firey theory. II: Affine and geominimal surface areas. Adv. Math. 118, 244-294 (1996) · Zbl 0853.52005 · doi:10.1006/aima.1996.0022 |
| [27] | Lutwak, E., Yang, D., Zhang, G.: Optimal Sobolev norms and the Lp-Minkowski problem. Int. Math. Res. Notices 2006 (2006). doi:10.1155/IMRN/2006/62987 · Zbl 1110.46023 |
| [28] | Lutwak, E., Yang, D., Zhang, G.: Sharp affine Lp Sobolev inequalities. J. Differ. Geom. 62, 17-38 (2002) · Zbl 1073.46027 |
| [29] | Lutwak, E., Yang, D., Zhang, G.: The Cramer-Rao inequality for star bodies. Duke Math. J. 112, 59-81 (2002) · Zbl 1021.52008 · doi:10.1215/S0012-9074-02-11212-5 |
| [30] | Lutwak, E., Yang, D., Zhang, G.: Lp affine isoperimetric inequalities. J. Differ. Geom. 56, 111-132 (2000) · Zbl 1034.52009 |
| [31] | Meyer, M., Werner, E.: On the p-affine surface area. Adv. Math. 152, 288-313 (2000) · Zbl 0964.52005 · doi:10.1006/aima.1999.1902 |
| [32] | Petty, C. M.; Goodman, J. E. (ed.); Lutwak, E. (ed.); Malkevitch, J. (ed.); Pollack, R. (ed.), Affine isoperimetric problems, discrete geometry and convexity, No. 440, 113-127 (1985) · Zbl 0576.52003 |
| [33] | Schütt, C., Werner, E.: Surface bodies and p-affine surface area. Adv. Math. 187, 98-145 (2004) · Zbl 1089.52002 · doi:10.1016/j.aim.2003.07.018 |
| [34] | Sapiro, G., Tannenbaum, A.: On affine plane curve evolution. J. Funct. Anal. 119, 79-120 (1994) · Zbl 0801.53008 · doi:10.1006/jfan.1994.1004 |
| [35] | Stancu, A.: The discrete planar L0-Minkowski problem. Adv. Math. 167, 160-174 (2002) · Zbl 1005.52002 · doi:10.1006/aima.2001.2040 |
| [36] | Stancu, A.: On the number of solutions to the discrete two-dimensional L0-Minkowski problem. Adv. Math. 180, 290-323 (2003) · Zbl 1054.52001 · doi:10.1016/S0001-8708(03)00005-7 |
| [37] | Stancu, A.: Two volume product inequalities and their applications. Can. Math. Bull. 52, 464-472 (2004) · Zbl 1178.52005 · doi:10.4153/CMB-2009-049-0 |
| [38] | Stancu, A.: The floating body problem. Bull. Lond. Math. Soc. 38, 839-846 (2006) · Zbl 1115.52002 · doi:10.1112/S0024609306018728 |
| [39] | Stancu, A.: The necessary condition for the discrete L0-Minkowski problem in ℝ2. J. Geom. 88, 162-168 (2008) · Zbl 1132.52300 · doi:10.1007/s00022-007-1937-4 |
| [40] | Stancu, A.: Centro-affine invariants for smooth convex bodies. Int. Math. Res. Not. (2011). doi:10.1093/imrn/rnr110 · Zbl 1250.52009 · doi:10.1093/imrn/rnr110 |
| [41] | Stancu, A.: Some affine invariants revisited. arXiv:1208.0783v1 [math.FA] · Zbl 1276.52011 |
| [42] | Trudinger, N.S., Wang, X.J.: The Bernstein problem for affine maximal hypersurfaces. Invent. Math. 140, 399-422 (2000) · Zbl 0978.53021 · doi:10.1007/s002220000059 |
| [43] | Trudinger, N.S., Wang, X.J.: Affine complete locally convex hypersurfaces. Invent. Math. 150, 45-60 (2002) · Zbl 1071.53008 · doi:10.1007/s00222-002-0229-8 |
| [44] | Trudinger, N.S., Wang, X.J.: The affine Plateau problem. J. Am. Math. Soc. 18, 253-289 (2005) · Zbl 1229.53049 · doi:10.1090/S0894-0347-05-00475-3 |
| [45] | Trudinger, N.S., Wang, X.J.: Boundary regularity for the Monge-Ampère and affine maximal surface equations. Ann. Math. 167, 993-1028 (2008) · Zbl 1176.35046 · doi:10.4007/annals.2008.167.993 |
| [46] | Werner, E., Ye, D.: New Lp affine isoperimetric inequalities. Adv. Math. 218, 762-780 (2008) · Zbl 1155.52002 · doi:10.1016/j.aim.2008.02.002 |
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.
