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Kateryna Tatarko：Isoperimetric problem from classical to reverse——II
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https://www.youtube.com/watch?v=V68uxxnz2o8 Hausdorff Center for Mathematics The well-known classical isoperimetric problem states that the Euclidean ball has the largest volume among all convex bodies in Rn of a fixed surface area. We will discuss the question of reversing this result for the class of convex bodies with curvature at each point of their boundary bounded below by some positive constant. In particular, we discuss the problem of minimizing the volume in this class of convex bodies and resolve it in R3. S. Myroshnychenko, K. Tatarko, V. Yaskin——“Answers to questions of Grünbaum and Loewner”：https://doi.org/10.48550/arXiv.2404.15188 We construct a convex body K in Rn, n≥5, with the property that there is exactly one hyperplane H passing through c(K), the centroid of K, such that the centroid of K∩H coincides with c(K). This provides answers to questions of Grünbaum and Loewner for n≥5. The proof is based on the existence of non-intersection bodies in these dimensions. Kostiantyn Drach, Kateryna Tatarko——“Reverse isoperimetric problems under curvature constraints”：https://doi.org/10.48550/arXiv.2303.02294 In this paper we solve several reverse isoperimetric problems in the class of λ-convex bodies, i.e., convex bodies whose curvature at each point of their boundary is bounded below by some λ>0. We give an affirmative answer in R3 to a conjecture due to Borisenko which states that the λ-convex lens, i.e., the intersection of two balls of radius 1/λ, is the unique minimizer of volume among all λ-convex bodies of given surface area. Also, we prove a reverse inradius inequality: in model spaces of constant curvature and arbitrary dimension, we show that the λ-convex lens (properly defined in non-zero curvature spaces) has the smallest inscribed ball among all λ-convex bodies of given surface area. This solves a conjecture due to Bezdek on minimal inradius of isoperimetric ball-polyhedra in Rn.
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