Abstract
The variation of the functional U of a convex body in \({\mathbb {R}}^n\) introduced by Lutwak–Yang–Zhang is derived. It becomes the first mixed volume of Minkowski when the convex body is strictly convex. A Minkowski-type inequality for the variation of the of U is proved, which implies the LYZ conjecture for the functional U directly.
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Ball, K.: Shadows of convex bodies. Trans. Am. Math. Soc. 327(2), 891–901 (1991)
Böröczky, K., Henk, M.: Cone-volume measure of general centered convex bodies. Adv. Math. 286, 703–721 (2016)
Böröczky, K.J., Lutwak, E., Yang, D., Zhang, G.: The logarithmic Minkowski problem. J. Am. Math. Soc. 26(3), 831–852 (2013)
Böröczky, K.J., Lutwak, E., Yang, D., Zhang, G.: Affine images of isotropic measures. J. Differ. Geom. 99(3), 407–442 (2015)
Gardner, R.J.: Geometric Tomography, 2nd edn. Encyclopedia of Mathematics and Its Applications, vol. 58. Cambridge University Press, Cambridge (2006)
Gardner, R.J., Zhang, G.: Affine inequalities and radial mean bodies. Am. J. Math. 120(3), 505–528 (1998)
Goodey, P., Zhang, G.Y.: Characterizations and inequalities for zonoids. J. Lond. Math. Soc. 53(1), 184–196 (1996)
Gordon, Y., Meyer, M., Reisner, S.: Zonoids with minimal volume-product—a new proof. Proc. Am. Math. Soc. 104(1), 273–276 (1988)
Gruber, P.M.: Convex and Discrete Geometry. Grundlehren der Mathematischen Wissenschaften, vol. 336. Springer, Berlin (2007)
He, B., Leng, G., Li, K.: Projection problems for symmetric polytopes. Adv. Math. 207(1), 73–90 (2006)
Henk, M., Linke, E.: Cone-volume measures of polytopes. Adv. Math. 253, 50–62 (2014)
Hu, J., Xiong, G.: A new affine invariant geometric functional for polytopes and its associated affine isoperimetric inequalities. Int. Math. Res. Not. IMRN. https://doi.org/10.1093/imrn/rnz090
Ludwig, M.: Projection bodies and valuations. Adv. Math. 172(2), 158–168 (2002)
Lutwak, E.: Mixed projection inequalities. Trans. Am. Math. Soc. 287(1), 91–105 (1985)
Lutwak, E.: Inequalities for mixed projection bodies. Trans. Am. Math. Soc. 339(2), 901–916 (1993)
Lutwak, E., Yang, D., Zhang, G.: A new affine invariant for polytopes and Schneider’s projection problem. Trans. Am. Math. Soc. 353(5), 1767–1779 (2001)
Petty, C.M.: Isoperimetric problems. In: Kay, D.C. (ed.) Proceedings of the Conference on Convexity and Combinatorial Geometry, pp. 26–41. University of Oklahoma, Norman (1971)
Schneider, R.: Random hyperplanes meeting a convex body. Z. Wahrsch. Verw. Gebiete 61(3), 379–387 (1982)
Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory, 2nd edn. Encyclopedia of Mathematics and Its Applications, vol. 151. Cambridge University Press, Cambridge (2014)
Schneider, R., Weil, W.: Zonoids and related topics. In: Gruber, P.M., Wills, J.M. (eds.) Convexity and Its Applications, pp. 296–317. Birkhäuser, Basel (1983)
Thompson, A.C.: Minkowski Geometry. Encyclopedia of Mathematics and Its Applications, vol. 63. Cambridge University Press, Cambridge (1996)
Xiong, G.: Extremum problems for the cone volume functional of convex polytopes. Adv. Math. 225(6), 3214–3228 (2010)
Zhang, G.Y.: Restricted chord projection and affine inequalities. Geom. Dedicata 39(2), 213–222 (1991)
Zou, D., Xiong, G.: The Orlicz Brunn–Minkowski inequality for the projection body. J. Geom. Anal. https://doi.org/10.1007/s12220-019-00182-7
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Lu, X., Sun, Q. & Xiong, G. A New Approach to the Minkowski First Mixed Volume and the LYZ Conjecture. Discrete Comput Geom 66, 122–139 (2021). https://doi.org/10.1007/s00454-019-00148-0
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DOI: https://doi.org/10.1007/s00454-019-00148-0