Extremum problems for the cone volume functional of convex polytopes

By Ge Xiong. 2010

93 citations recorded by the source corpus.

Abstract Lutwak, Yang and Zhang defined the cone volume functional U over convex polytopes in R n containing the origin in their interiors, and conjectured that the greatest lower bound on the ratio of this centro-affine invariant U to volume V is attained by parallelotopes. In this paper, we give affirmative answers to the conjecture in R 2 and R 3 . Some new sharp inequalities characterizing parallelotopes in R n are established. Moreover, a simplified proof for the conjecture restricted to the class of origin-symmetric convex polytopes in R n is provided.

Subfields: Applied Mathematics, Pharmacology

View Paper: https://www.semanticscholar.org/paper/d0fee38904e0111e1e0624265fec0bbe39c862ee

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In Applied Mathematics

1. Yiming Zhao
2. Yong Huang
3. K. Boroczky
4. Deane Yang
5. Gaoyong Zhang
6. Erwin Lutwak
7. Deping Ye
8. Dongmeng Xi

In Pharmacology

1. Yiming Zhao
2. Deane Yang
3. Gaoyong Zhang
4. Erwin Lutwak
5. Gangsong Leng
6. K. Boroczky
7. Yong Huang
8. Martin Henk

Citations in the corpus

• The logarithmic Minkowski problem (2012, unknown): “For origin-symmetric polytopes, the subspace concentration inequality of Theorem 5.2 was established previously by He, Leng and Li [28], with an alternate proof given by Xiong [62].”
• Dar’s conjecture and the log–Brunn–Minkowski inequality (2016, background): “It was first proved by He, Leng, and Li [27] that the cone-volume measures of o-symmetric polytopes in Rn satisfy the subspace concentration inequality (2.1), see Xiong [38] for an alternate proof.”
• Existence of solutions to the even dual Minkowski problem (2018, background): “Naor [47], Naor & Romik [48], Paouris & Werner [51], Stancu [61, 62], Xiong [65], Zhu [70], and Zou & Xiong [74].”
• Cone-volume measures of polytopes (2013, methodology): “3) was proven in the special case dimL = 1 by Hernández Cifre (2007, private communication), in the cases dimL = 1, n− 1 by Xiong [27], and for originsymmetric polytopes independently by Henk, Schürmann & Wills [14] and by He, Leng & Li [13].”
• The logarithmic Minkowski problem for polytopes (2014, unknown): “In [6], Böröczky, Lutwak, Yang and Zhang (for polytopes see He, Leng and Li [31], and Xiong [71]) showed that if ξ is a subspace of Rn with 0 dim ξ n and µ is the cone-volume measure of an origin-symmetric convex body, then µ(ξ ∩ Sn−1) ≤ dim ξ n µ(Sn−1).”
• On the L dual Minkowski problem (2018, background): “See, for example, Barthe, Guédon, Mendelson & Naor [4], Böröczky & Henk [7], Böröczky, Lutwak, Yang & Zhang [9–11], Henk & Linke [29], Ludwig [44], Ludwig & Reitzner [47], Naor [54], Naor & Romik [55], Paouris & Werner [56], Stancu [62,63], Xiong [65], Zhu [70], and Zou & Xiong [73].”
• Affine images of isotropic measures (2015, background): “[77] G. Xiong, Extremum problems for the cone volume functional of convex polytopes, Adv.”
• Cone-volume measure of general centered convex bodies (2016, unknown): “Xiong [44], respectively.”
• The Minkowski problem in Gaussian probability space (2020, background): “Due to limit of space, we mention only a selection of beautiful results in this area, [10,12,33,34,54,56,60,61,63,65, 69,70,72] and refer the interested readers to Schneider’s book [66] for more details.”