9 papers
Tight Stability Estimates Near the Simplex and Applications for the Banach-Mazur Distance and Rogers-Shephard-Type Inequalities
René Brandenberg, Bernardo González Merino, Florian Grundbacher
We establish a tight stability estimate for the Minkowski asymmetry near its maximal value, improving earlier results in both the range for the admissible error and the strength of…
Exact Banach-Mazur distances of certain -sums and cones
Florian Grundbacher, Tomasz Kobos
We determine certain Banach-Mazur distances involving -direct sums of finite-dimensional real normed spaces and related cone constructions of convex bodies. Using a recent…
John-type decompositions for affinely-optimal positions of convex bodies
Florian Grundbacher, Tomasz Kobos
Many classical problems in convex geometry can be cast as optimization problems under certain containment conditions. The arguably best-understood example is volume-maximization of…
On the minimal area of quadrangles circumscribed about planar convex bodies
Ferenc Fodor, Florian Grundbacher
We show that every planar convex body is contained in a quadrangle whose area is less than times the area of the original convex body, improving…
Minkowski chirality: a measure of reflectional asymmetry of convex bodies
Andrei Caragea, Katherina von Dichter, Kurt Klement Gottwald +3
Using an optimal containment approach, we quantify the asymmetry of convex bodies in with respect to reflections across affine subspaces of a given dimension. We pro…
Large Signed Sums and the Polarization Constant of Convex Bodies
Gergely Ambrus, Florian Grundbacher
As a counterpart to classical vector balancing, we study large signed sums of unit vectors in finite-dimensional Minkowski spaces and develop their connection with polarization pro…
Tightening Inequalities on Volume Extremal -Ellipsoids Using Asymmetry Measures
René Brandenberg, Florian Grundbacher
We consider two well-known problems: upper bounding the volume of lower dimensional ellipsoids contained in convex bodies given their John ellipsoid, and lower bounding the volume…
On Certain Extremal Banach-Mazur Distances and Ader's Characterization of Distance Ellipsoids
Florian Grundbacher, Tomasz Kobos
A classical consequence of the John Ellipsoid Theorem is the upper bound on the Banach-Mazur distance between the Euclidean ball and any symmetric convex body in $\mathb…
-Means of Convex Bodies: Sharpening Relations and Structural Properties
René Brandenberg, Florian Grundbacher
We study general -means of convex bodies, extending the classical definitions by W. J. Firey via support and gauge functions to two families ranging over all $p \in [-\infty,\in…