Inradius Estimates for Convex Domains in 2-Dimensional Alexandrov Spaces

Drach, Kostiantyn

doi:10.1515/agms-2018-0009

Cited by 3 publications

(1 citation statement)

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“…For 2-dimensional λ-convex bodies in the Euclidean plane, i.e., the space of constant zero curvature, Milka [Mi2] proved that the λ-convex lens has the smallest inradius for a given length of the boundary. Much later, this result was generalized in [Dr4] to arbitrary λ-convex domains in 2-dimensional Alexandrov spaces of curvature bounded below, thus closing this question in dimension 2. Recently, Bezdek [Be1] showed that among all λconvex bodies in R n , n 2, the λ-convex lens has the smallest inradius for a given volume.…”

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confidence: 98%

Reverse isoperimetric problems under curvature constraints

Drach¹,

Tatarko²

2023

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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 R 3 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 R n .

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confidence: 98%