On volume product inequalities for convex sets
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- by Stefano Campi and Paolo Gronchi
- Proc. Amer. Math. Soc. 134 (2006), 2393-2402
- DOI: https://doi.org/10.1090/S0002-9939-06-08241-4
- Published electronically: February 3, 2006
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Abstract:
The volume of the polar body of a symmetric convex set $K$ of ${\mathbb {R}^d}$ is investigated. It is shown that its reciprocal is a convex function of the time $t$ along movements, in which every point of $K$ moves with constant speed parallel to a fixed direction. This result is applied to find reverse forms of the $L^{p}$-Blaschke-Santaló inequality for two-dimensional convex sets.References
- A. D. Aleksandrov, A general method of majorizing solutions of the Dirichlet problem, Sibirsk. Mat. Ž. 7 (1966), 486–498 (Russian). MR 0217421
- A. D. Aleksandrov, Mean values of the support function, Dokl. Akad. Nauk SSSR 172 (1967), 755–758 (Russian). MR 0212681
- J. Bourgain and V. D. Milman, New volume ratio properties for convex symmetric bodies in $\textbf {R}^n$, Invent. Math. 88 (1987), no. 2, 319–340. MR 880954, DOI 10.1007/BF01388911
- Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285, Springer-Verlag, Berlin, 1988. Translated from the Russian by A. B. Sosinskiĭ; Springer Series in Soviet Mathematics. MR 936419, DOI 10.1007/978-3-662-07441-1
- Stefano Campi, Andrea Colesanti, and Paolo Gronchi, A note on Sylvester’s problem for random polytopes in a convex body, Rend. Istit. Mat. Univ. Trieste 31 (1999), no. 1-2, 79–94. MR 1763244
- S. Campi and P. Gronchi, The $L^p$-Busemann-Petty centroid inequality, Adv. Math. 167 (2002), no. 1, 128–141. MR 1901248, DOI 10.1006/aima.2001.2036
- Stefano Campi and Paolo Gronchi, On the reverse $L^p$-Busemann-Petty centroid inequality, Mathematika 49 (2002), no. 1-2, 1–11 (2004). MR 2059037, DOI 10.1112/S0025579300016004
- Richard J. Gardner, Geometric tomography, Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, Cambridge, 1995. MR 1356221
- R. J. Gardner, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 3, 355–405. MR 1898210, DOI 10.1090/S0273-0979-02-00941-2
- 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. MR 958082, DOI 10.1090/S0002-9939-1988-0958082-9
- Erwin Lutwak, Selected affine isoperimetric inequalities, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 151–176. MR 1242979, DOI 10.1016/B978-0-444-89596-7.50010-9
- Erwin Lutwak, Deane Yang, and Gaoyong Zhang, $L_p$ affine isoperimetric inequalities, J. Differential Geom. 56 (2000), no. 1, 111–132. MR 1863023
- Erwin Lutwak and Gaoyong Zhang, Blaschke-Santaló inequalities, J. Differential Geom. 47 (1997), no. 1, 1–16. MR 1601426
- Kurt Mahler, Ein Übertragungsprinzip für konvexe Körper, Časopis Pěst. Mat. Fys. 68 (1939), 93–102 (German). MR 0001242
- K. Mahler, Ein Minimalproblem für konvexe Polygone, Mathematica (Zutphen) B 7 (1939), 118–127.
- Mathieu Meyer, Convex bodies with minimal volume product in $\textbf {R}^2$, Monatsh. Math. 112 (1991), no. 4, 297–301. MR 1141097, DOI 10.1007/BF01351770
- Mathieu Meyer and Alain Pajor, On the Blaschke-Santaló inequality, Arch. Math. (Basel) 55 (1990), no. 1, 82–93. MR 1059519, DOI 10.1007/BF01199119
- Shlomo Reisner, Random polytopes and the volume-product of symmetric convex bodies, Math. Scand. 57 (1985), no. 2, 386–392. MR 832364, DOI 10.7146/math.scand.a-12124
- Shlomo Reisner, Zonoids with minimal volume-product, Math. Z. 192 (1986), no. 3, 339–346. MR 845207, DOI 10.1007/BF01164009
- C. A. Rogers and G. C. Shephard, Some extremal problems for convex bodies, Mathematika 5 (1958), 93–102. MR 104203, DOI 10.1112/S0025579300001418
- 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. Univ. Pierre et Marie Curie, vol. 46, Univ. Paris VI, Paris, 1981, pp. Exp. No. 11, 25 (French). MR 670798
- Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521, DOI 10.1017/CBO9780511526282
- G. C. Shephard, Shadow systems of convex sets, Israel J. Math. 2 (1964), 229–236. MR 179686, DOI 10.1007/BF02759738
- Giorgio Talenti, Some estimates of solutions to Monge-Ampère type equations in dimension two, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), no. 2, 183–230. MR 623935
Bibliographic Information
- Stefano Campi
- Affiliation: Dipartimento di Matematica Pura e Applicata “G. Vitali", Università degli Studi di Modena e Reggio Emilia, Via Campi 213/b, 41100 Modena, Italy
- Address at time of publication: Dipartimento di Ingegneria dell’Informazione, Università degli Studi di Siena, Via Roma 56, 53100 Siena, Italy
- MR Author ID: 205850
- Email: Campi@unimo.it, Campi@dii.unisi.it
- Paolo Gronchi
- Affiliation: Istituto per le Applicazioni del Calcolo - Sezione di Firenze, Consiglio Nazionale delle Ricerche Via Madonna del Piano, Edificio F, 50019 Sesto Fiorentino (FI), Italy
- Address at time of publication: Dipartimento di Matematica e Applicazioni per l’Architettura, Università degli Studi di Firenze, Piazza Ghiberti 27, 50122 Firenze, Italy
- MR Author ID: 340283
- Email: P.Gronchi@fi.iac.cnr.it, P.Gronchi@fi.iac.cnr.it
- Received by editor(s): July 27, 2004
- Received by editor(s) in revised form: March 4, 2005
- Published electronically: February 3, 2006
- Communicated by: N. Tomczak-Jaegermann
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 2393-2402
- MSC (2000): Primary 52A40
- DOI: https://doi.org/10.1090/S0002-9939-06-08241-4
- MathSciNet review: 2213713