arXiv research

Defines circumcenter of mass for polytopes without triangulation.

Study circumcenters in Finsler unitary groups with optimal convexity bounds.

Given a Moebius homeomorphism $f : \partial X \to \partial Y$ between boundaries of proper, geodesically complete CAT(-1) spaces $X,Y$, we describe an extension $\hat{f} : X \to Y$ of $f$, called the circumcenter map of $f$, which is constructed using circumcenters of expanding sets. The extension $\hat{f}$ is shown to…

Study on Poncelet polygons' centers and circumcenters in various geometries.

We show that every cross ratio preserving homeomorphism between boundaries of Hadamard manifolds extends to a continuous map, called circumcenter extension, provided that the manifolds satisfy certain visibility conditions. We show that this map is a rough isometry, whenever the manifolds admit cocompact group actions …

Given a Moebius homeomorphism $f : \partial X \to \partial Y$ between boundaries of proper, geodesically complete CAT(-1) spaces $X,Y$, and a family of probability measures $\{ μ_x \}_{x \in X}$ on $\partial X$, we describe a continuous family of extensions $\{\hat{f}_p : X \to Y \}_{1 \leq p \leq \infty}$ of $f$, call…

We introduce a proximal subdifferential and develop a calculus for nonsmooth functions defined on any Riemannian manifold $M$. We give several applications of this theory, concerning: 1) differentiability and geometrical properties of the distance function to a closed subset $C$ of $M$; 2) solvability and implicit func…

We study iterations of two classical constructions, the evolutes and involutes of plane curves, and we describe the limiting behavior of both constructions on a class of smooth curves with singularities given by their support functions. Next we study two kinds of discretizations of these constructions: the curves are r…