arXiv subjects

Filip Fryš

Publications and source records attributed to Filip Fryš.

4 recordsLinked to original sources

arXiv2026

Around higher-order Godbersen conjectures

We consider two higher-order generalizations of the Godbersen conjecture for mixed volumes of convex bodies, which were first proposed by Schneider in 2000. First, we establish the conjectures in several special cases, in particular in low dimensions. Second, we prove that certain consequences of the conjectures hold. More precisely, we define a higher-order version of the unbalanced difference body and prove related weighted inequalities, generalizing previous results of Artstein-Avidan and Putterman. Finally, we introduce higher-order analogs of unbalanced joins of convex bodies considered by Artstein-Avidan, Einhorn, Florentin, and Ostrover, prove the corresponding volume bounds, and propose conjectures that interpolate between the higher-order Godbersen conjectures and a conjecture due to F\'ary and R\'edei.

math.MG↗

arXiv2026

Anisotropic Minkowski Content for Countably $\mathcal{H}^k$-rectifiable Sets

We study anisotropic Minkowski content for lower-dimensional rectifiable sets. First, we prove that, for every convex body $C\subseteq\mathbb R^n$, the $k$-dimensional $C$-anisotropic Minkowski content of every compact $k$-rectifiable set exists. We show that it is given by an integral involving the $(n-k)$-dimensional volumes of the projections of $C$ onto the approximate normal spaces of the set. We then establish the same formula for closed countably $\mathcal H^k$-rectifiable sets of finite $\mathcal H^k$-measure satisfying an AFP-$k$-condition relative to the linear span of $C$, provided that the condition is witnessed by a finite Radon measure. Finally, we prove that, for a countably $\mathcal H^k$-rectifiable set, if the formula holds for one full-dimensional convex body, then it holds for every full-dimensional convex body.

math.CA↗

arXiv2025

Anisotropic lower-dimensional Minkowski content and $\mathcal{S}$-content

This paper investigates the lower-dimensional anisotropic Minkowski content and $\mathcal{S}$-content. We establish that these anisotropic contents exhibit properties analogous to their isotropic counterparts by proving analogous inequalities between the lower-dimensional anisotropic Minkowski content and $\mathcal{S}$-content. A key component of our approach is demonstrating that the associated anisotropic volume function is of Kneser type, a result that underpins many of our proofs. In addition, we introduce anisotropic versions of the Minkowski and $\mathcal{S}$-dimensions and derive inequalities relating them. As an application, we analyze the existence of the $\log_2(3)$-dimensional anisotropic Minkowski and $\mathcal{S}$-contents of the Sierpinski gasket.

math.CA↗

arXiv2025

Existence of Anisotropic Minkowski Content

This paper is devoted to the existence of anisotropic Minkowski content and anisotropic outer Minkowski content. Our result is that the Minkowski content of the topological boundary of a given set of finite perimeter $E$ coincides with the perimeter of $E$ if and only if the anisotropic Minkowski content of the topological boundary of $E$ coincides with half of the sum of the anisotropic perimeter of $E$ and the anisotropic perimeter of the complement of $E.$ As a consequence, we find that the existence of anisotropic outer Minkowski content of a given set of finite perimeter and its complement ensures the existence of outer Minkowski content of the set and its complement.

math.CA↗