Abstract
The optimal condition of the cone volume measure of a pair of antipodal points is proved and analyzed.
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F. Barthe, O. Guedon, S. Mendelson and A. Naor, A probabilistic approach to the geometry of the \({l_{p}^{n}}\)-ball, Ann. of Probability, 33 (2005), 480–513.
K. J. Böröczky, P. Hegedűs and G. Zhu, The discrete logarithmic Minkowski problem, IMRN, submitted. arXiv:1409.7907, 4fu9n
K. J. Böröczky and M. Henk, Cone-volume measure and stability, arXiv:1407.7272, submitted.
K.J. Böröczky, E. Lutwak, D. Yang and G. Zhang, The log-Brunn–Minkowski inequality, Adv. Math., 231 (2012), 1974–1997.
K. J. Böröczky, E. Lutwak, D. Yang and G. Zhang, The logarithmic Minkowski problem, J. Amer. Math. Soc., 26 (2013), 831–852.
M. Gromov and V. D. Milman, Generalization of the spherical isoperimetric inequality for uniformly convex Banach Spaces, Compositio Math., 62 (1987), 263–282.
P. M. Gruber, Convex and Discrete Geometry, Grundlehren der Mathematischen Wissenschaften, 336, Springer (Berlin, 2007).
He B, Leng G, Li K: Projection problems for symmetric polytopes. Adv. Math. 207, 73–90 (2006)
Henk M, Schürmann A, Wills J. M: Ehrhart polynomials and successive minima. Mathematika 52, 1–16 (2005)
Ludwig M.: General affine surface areas. Adv. Math. 224, 2346–2360 (2010)
M. Ludwig and M. Reitzner, A classification of SL(n) invariant valuations, Ann. of Math., 172 (2010), 1223–1271.
E. Lutwak, The Brunn–Minkowski–Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom., 38 (1993), 131–150, MR 1231704, Zbl 0788.52007.
E. Lutwak, D. Yang and G. Zhang, L p John ellipsoids, Proc. London Math. Soc., 90 (2005), 497–520.
A. Naor, The surface measure and cone measure on the sphere of \({l_{p}^{n}}\), Trans. Amer. Math. Soc., 359 (2007), 1045–1079.
A. Naor and D. Romik, Projecting the surface measure of the sphere of \({l_{p}^{n}}\), Ann. Inst. H. Poincaré Probab. Statist., 39 (2003), 241–261.
G. Paouris and E. Werner, Relative entropy of cone measures and L p centroid bodies, Proc. London Math. Soc., 104 (2012), 253–286.
R. Schneider, Convex Bodies: the Brunn–Minkowski Theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press (Cambridge, 1993), Second expanded edition, 2014.
A. Stancu, The discrete planar L 0-Minkowski problem, Adv. Math., 167 (2002), 160–174.
A. Stancu, On the number of solutions to the discrete two-dimensional L 0-Minkowski problem, Adv. Math., 180 (2003), 290–323.
A. Stancu, Centro-affine invariants for smooth convex bodies, Int. Math. Res. Not., (2012), 2289–2320.
A. C. Thompson, Minkowski Geometry, Encyclopedia of Mathematics and its Applications, vol. 63, Cambridge University Press (Cambridge, 1996).
Zhu G.: The logarithmic Minkowski problem for polytopes. Adv. Math. 262, 909–931 (2014)
G. Zhu, The centro-affine Minkowski problem for polytopes, accepted for publication in J. Differential Geom.
G. Zhu, The L p Minkowski problem for polytopes, arXiv:1406.7503.
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Böröczky, K.J., Hegedűs, P. The cone volume measure of antipodal points. Acta Math. Hungar. 146, 449–465 (2015). https://doi.org/10.1007/s10474-015-0511-z
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DOI: https://doi.org/10.1007/s10474-015-0511-z