arXiv subjects

Katherina von Dichter

Publications and source records attributed to Katherina von Dichter.

7 recordsLinked to original sources

arXiv2026

On the Log-submodularity for zonoids: from Mixed Volume inequalities to the Hypercube

We prove a log-submodularity-type inequality for zonoids in $\mathbb{R}^4$, extending the three-dimensional result of Fradelizi, Madiman, Meyer, and Zvavitch. More generally, we conjecture a log-submodularity-type inequality for zonoids in arbitrary dimension. This inequality admits several equivalent formulations in terms of volumes of coordinate projections as well as in terms of mixed volumes, thereby unifying several geometric perspectives. We reduce the conjectured inequality to a polynomial inequality whose variables are associated with the vertices of a hypercube and whose coefficients encode the volumes of 0/1 simplices. This reduction reveals unexpected connections between mixed volumes of zonoids, matroid theory, and real algebraic geometry.

math.MG↗

arXiv2026

Mixed volumes of zonoids and the absolute value of the Grassmannian

Zonoids are Hausdorff limits of zonotopes, while zonotopes are convex polytopes defined as the Minkowski sums of finitely many segments. We present a combinatorial framework that links the study of mixed volumes of zonoids (a topic that has applications in algebraic combinatorics) with the study of the absolute value of the Grassmannian, defined as the image of the Grassmannian under the coordinate-wise absolute value map. We use polyhedral computations to derive new families of inequalities for n zonoids in dimension d, when (n,d)=(6,2) and (6,3). Unlike the classical geometric inequalities, originating from the Brunn-Minkowski and Aleksandrov-Fenchel inequalities, the inequalities we produce have the special feature of being Minkowski linear in each of the n zonoids they involve.

math.CO↗

arXiv2025

Bounding the diameter-width ratio using containment inequalities of means of convex bodies

We completely describe the region of possible values of the diameter-width ratio for planar pseudo-complete sets in dependence of the Minkowski asymmetry. In order to do this, we focus on the containment inequalities of $K \cap (-K)$ and $\frac{K-K}{2}$ for a Minkowski centered convex compact set $K$, i.e. we define $τ(K)$ to be the smallest possible factor to cover $K \cap (-K)$ by a rescalation of $\frac{K-K}{2}$ and give the region of the possible values of $τ(K)$ in the planar case in dependence of the Minkowski asymmetry of $K$.

math.MG↗

arXiv2025

Minkowski chirality: a measure of reflectional asymmetry of convex bodies

Using an optimal containment approach, we quantify the asymmetry of convex bodies in $\mathbb{R}^n$ with respect to reflections across affine subspaces of a given dimension. We prove general inequalities relating these ''Minkowski chirality'' measures to Banach--Mazur distances and to each other, and prove their continuity with respect to the Hausdorff distance. In the planar case, we determine the reflection axes at which the Minkowski chirality of triangles and parallelograms is attained, and show that $\sqrt{2}$ is a tight upper bound on the chirality in both cases.

math.MG↗

arXiv2024

From inequalities relating symmetrizations of convex bodies to the diameter-width ratio for complete and pseudo-complete convex sets

For a Minkowski centered convex compact set $K$ we define $α(K)$ to be the smallest possible factor to cover $K \cap (-K)$ by a rescalation of $\mathrm{conv} (K\cup (-K))$ and give a complete description of the possible values of $α(K)$ in the planar case in dependence of the Minkowski asymmetry of $K$. As a side product, we show that, if the asymmetry of $K$ is greater than the golden ratio, the boundary of $K$ intersects the boundary of its negative $-K$ always in exactly 6 points. As an application, we derive bounds for the diameter-width-ratio for pseudo-complete and complete sets, again in dependence of the Minkowski asymmetry of the convex bodies, tightening those depending solely on the dimension given in a recent result of Richter [10].

math.MG↗

arXiv2022

Relating Symmetrizations of Convex Bodies: Once More the Golden Ratio

We show that for any Minkowski centered planar convex compact set $C$ the Harmonic mean of $C$ and $-C$ can be optimally contained in the arithmetic mean of the same sets if and only if the Minkowski asymmetry of $C$ is at most the golden ratio $(1+\sqrt{5})/2 \approx 1.618$. Moreover, the most asymmetric such set that is (up to a linear transformation) a special pentagon, which we call the golden house.

math.MG↗

arXiv2021

Tightening and reversing the arithmetic-harmonic mean inequality for symmetrizations of convex sets

This paper deals with four symmetrizations of a convex set $C$: the intersection, the harmonic and the arithmetic mean, and the convex hull of $C$ and $-C$. A well-known result of Firey shows that those means build up a subset-chain in the given order. On the one hand, we determine the dilatation factors, depending on the asymmetry of $C$, to reverse the containments between any of those symmetrizations. On the other hand, we tighten the relations proven by Firey and show a stability result concerning those factors near the simplex.

math.MG↗