Extremum problems for the cone volume functional of convex polytopes

Affine images of isotropic measures

“…For polytopes, the inequalitypart of the subspace concentration condition of Lemma 8.3 was established by He, Leng, and Li [30], with a shorter proof provided by Xiong [77]. Theorem 8.4.…”

“…When K is an origin-symmetric polytope, He, Leng, and Li [30] established inequality (8.6), and later Xiong [77] gave a simplified proof. Xiong [77] proved (8.6) for polytopes in two and three dimensions.…”

“…Xiong [77] proved (8.6) for polytopes in two and three dimensions. Here, we establish (8.6) under a condition.…”

“…For two and three dimensional polytopes, an affirmative answer to Problem 8.9 has been given by Xiong [77].…”

On the approximation of a convex body by its radial mean bodies

“…We refer to e.g. [5,8,10,12,13,14,15,16,17,18,19,23,24,25] for further details, extensions and applications. As an aside, we observe that throughout the whole paper [8], all affine inequalities attain extremum if and only if the convex body is a simplex.…”