Page 139 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 139
CONVEX BODIES - APPROXIMATION AND SECTIONS (MS-61)
The Lp Minkowski problem and polytopal approximation
Karoly Boroczky, boroczky.karoly.j@renyi.hu
Alfred Renyi Institute of Mathematics, Hungary
The Lp Minkowski problem , a Monge-Ampere type equation on the sphere, is a recent version
of the classical Minkowski problem. I will review cases when one can reduce the equation to
properties of polytopes, and then use polytopal approximation to solve the PDE in general.
Strengthened inequalities for the mean width
Ferenc Fodor, fodorf@math.u-szeged.hu
University of Szeged, Hungary
According to a result of Barthe the regular simplex maximizes the mean width of convex bodies
whose John ellipsoid is the Euclidean unit ball. The reverse statement that the regular simplex
minimizes the mean width of convex bodies whose Löwner ellipsoid is the Euclidean unit ball is
also true as proved by Schmuckenschläger. In this talk we prove strengthened stability versions
of these theorems and also some related stability statements for the convex hull of the sup-
port of centered isotropic measures on the sphere. This is joint work with Károly J. Böröczky
(Budapest, Hungary) and Daniel Hug (Karlsruhe, Germany).
Colorful Helly-type Theorems for Ellipsoids
Viktória Földvári, foldvari@math.elte.hu
Alfréd Rényi Institute of Mathematics, Hungary
Coauthors: Gábor Damásdi, Márton Naszódi
Helly-type theorems have been widely studied and applied in discrete geometry. In this talk, I
am going to give a brief overview of quantitative Helly-type theorems and introduce our joint
result with Gábor Damásdi and Márton Naszódi, a colorful Helly-type theorem for ellipsoids.
Coverings by homothets of a convex body
Nora Frankl, nfrankl@andrew.cmu.edu
Carnegie Mellon University, United States
Coauthors: Janos Nagy, Marton Naszodi
Rogers proved that for any convex body K, we can cover Rd by translates of K of density
roughly d log d. We discuss several related results. First, we extend Roger’s result by showing
that, if we are given a family of positive homothets of K of infinite total volume, then we can
find appropriate translation vectors for each given homothet to cover Rd with the same density.
Second, we consider an extension to multiple coverings of space by translates of a convex body.
Finally, we also prove a lower bound on the total volume of a family F of homothets of K that
guarantees the existence of a covering of K by members of F.
137
The Lp Minkowski problem and polytopal approximation
Karoly Boroczky, boroczky.karoly.j@renyi.hu
Alfred Renyi Institute of Mathematics, Hungary
The Lp Minkowski problem , a Monge-Ampere type equation on the sphere, is a recent version
of the classical Minkowski problem. I will review cases when one can reduce the equation to
properties of polytopes, and then use polytopal approximation to solve the PDE in general.
Strengthened inequalities for the mean width
Ferenc Fodor, fodorf@math.u-szeged.hu
University of Szeged, Hungary
According to a result of Barthe the regular simplex maximizes the mean width of convex bodies
whose John ellipsoid is the Euclidean unit ball. The reverse statement that the regular simplex
minimizes the mean width of convex bodies whose Löwner ellipsoid is the Euclidean unit ball is
also true as proved by Schmuckenschläger. In this talk we prove strengthened stability versions
of these theorems and also some related stability statements for the convex hull of the sup-
port of centered isotropic measures on the sphere. This is joint work with Károly J. Böröczky
(Budapest, Hungary) and Daniel Hug (Karlsruhe, Germany).
Colorful Helly-type Theorems for Ellipsoids
Viktória Földvári, foldvari@math.elte.hu
Alfréd Rényi Institute of Mathematics, Hungary
Coauthors: Gábor Damásdi, Márton Naszódi
Helly-type theorems have been widely studied and applied in discrete geometry. In this talk, I
am going to give a brief overview of quantitative Helly-type theorems and introduce our joint
result with Gábor Damásdi and Márton Naszódi, a colorful Helly-type theorem for ellipsoids.
Coverings by homothets of a convex body
Nora Frankl, nfrankl@andrew.cmu.edu
Carnegie Mellon University, United States
Coauthors: Janos Nagy, Marton Naszodi
Rogers proved that for any convex body K, we can cover Rd by translates of K of density
roughly d log d. We discuss several related results. First, we extend Roger’s result by showing
that, if we are given a family of positive homothets of K of infinite total volume, then we can
find appropriate translation vectors for each given homothet to cover Rd with the same density.
Second, we consider an extension to multiple coverings of space by translates of a convex body.
Finally, we also prove a lower bound on the total volume of a family F of homothets of K that
guarantees the existence of a covering of K by members of F.
137
