Abstract
A variational formula for the Lutwak affine surface areas \(\Lambda _{j}\) of convex bodies in \(\mathbb {R}^n\) is established when \(1\le j\le n-1.\) By using introduced new ellipsoids associated with projection functions of convex bodies, we prove a sharp isoperimetric inequality for \(\Lambda _{j}\), which opens up a new passage to attack the longstanding Lutwak conjecture in convex geometry.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ball, K.: Volume ratios and a reverse isoperimetric inequality. J. Lond. Math. Soc. 44, 351–359 (1991)
Dann, S., Paouris, G., Pivovarov, P.: Bounding marginal densities via affine isoperimetry. Proc. Lond. Math. Soc. 113, 140–162 (2016)
Furstenberg, H., Tzkoni, I.: Spherical functions and integral geometry. Israel J. Math. 10, 327–338 (1971)
Gardner, R.: The Brunn–Minkowski inequality. Bull. Am. Math. Soc. 39, 355–405 (2002)
Gardner, R.: Geometric Tomography. Cambridge University Press, New York (2006)
Giannopoulos, A., Papadimitrakis, M.: Isotropic surface area measures. Mathematika 46, 1–13 (1999)
Grinberg, E.: Isoperimetric inequalities and identities for \(k\)-dimensional cross-sections of convex bodies. Math. Ann. 291, 75–86 (1991)
Gruber, P.: Convex and Discrete Geometry. Springer, Berlin (2007)
Hu, J., Xiong, G., Zou, D.: On mixed \(L_p\) John ellipsoids, Adv. Geom. (2019, to appear)
Hu, J., Xiong, G., Zou, D.: New affine inequalities and intersection mean ellipsoids (submitted)
Lutwak, E.: A general isepiphanic inequality. Proc. Am. Math. Soc. 90, 415–421 (1984)
Lutwak, E.: Inequalities for Hadwiger’s harmonic quermassintegrals. Math. Ann. 280, 165–175 (1988)
Lutwak, E.: Selected affine isoperimetric inequalities. In: Gruber, P.M., Wills, J.M. (eds.) Handbook of Convex Geometry, pp. 151–176. North-Holland, Amsterdam (1993)
Lutwak, E., Yang, D., Zhang, G.: A new ellipsoid associated with convex bodies. Duke Math. J. 104, 375–390 (2000)
Lutwak, E., Yang, D., Zhang, G.: Volume inequalities for subspaces of \(L_{p}\). J. Differ. Geom. 68, 159–184 (2004)
Lutwak, E., Yang, D., Zhang, G.: \(L_p\) John ellipsoids. Proc. Lond. Math. Soc. 90, 497–520 (2005)
Lutwak, E., Yang, D., Zhang, G.: Volume inequalities for isotropic measures. Am. J. Math. 129, 1711–1723 (2007)
Lutwak, E., Yang, D., Zhang, G.: A volume inequality for polar bodies. J. Differ. Geom. 84, 163–178 (2010)
Milman, V., Pajor, A.: Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed \(n\)-dimensional space. In: Lindenstrauss J., Milman V. D. (eds.) Geometric Aspects of Functional Analysis (1987–1988), Lecture Notes in Math., 1376, pp. 64–104. Springer, Berlin (1989)
Ossermann, R.: The isoperimetric inequality. Bull. Am. Math. Soc. 84, 1182–1238 (1978)
Petty, C.: Surface area of a convex body under affine transformations. Proc. Am. Math. Soc. 12, 824–828 (1961)
Petty, C.: Isoperimetric problems. In: Proceedings of the Conference on Convexity and Combinatorial Geometry, pp. 26-41. University of Oklahoma, Norman (1971)
Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory. Cambridge University Press, Cambridge (2014)
Schuster, F., Weberndorfer, M.: Volume inequalities for asymmetric Wulff shapes. J. Differ. Geom. 92, 263–283 (2012)
Xiong, G.: Extremum problems for the cone volume functional of convex polytopes. Adv. Math. 225, 3214–3228 (2010)
Zhang, G.: Restricted chord projection and affine inequalities. Geom. Dedicata 39, 213–222 (1991)
Zhang, G.: New affine isoperimetric inequalities. ICCM 265, 239–267 (2007)
Zou, D., Xiong, G.: Orlicz–John ellipsoids. Adv. Math. 265, 132–168 (2014)
Zou, D., Xiong, G.: Orlicz–Legendre ellipsoids. J. Geom. Anal. 26, 2474–2502 (2016)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Chang.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Research of the authors was supported by NSFC Nos. 11601399 and 11871373.
