Extremum problems for the cone volume functional of convex polytopes

Xiong Ge Tongji University

Convex and Discrete Geometry mathscidoc:1702.40002

Advances in Mathematics, 225, (6), 3214–3228, 2010.12
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 Rn are established. Moreover, a simplified proof for the conjecture restricted to the class of origin-symmetric convex polytopes in Rn is provided.
Convex polytope; Parallelotope; Centro-affine invariant; Cone volume functional; Projection body
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  title={Extremum problems for the cone volume functional of convex polytopes},
  author={Xiong Ge},
  booktitle={Advances in Mathematics},
Xiong Ge. Extremum problems for the cone volume functional of convex polytopes. 2010. Vol. 225. In Advances in Mathematics. pp.3214–3228. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170228123536435408560.
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