Geometry of simplices in Minkowski spaces

TL;DR

This paper explores the extension of classical Euclidean simplex properties to finite-dimensional normed spaces, introducing new generalizations of circumcenters, Euler lines, and Feuerbach spheres, along with duality-based theorems.

Contribution

It provides the first comprehensive generalizations of Euclidean simplex properties in Banach spaces, including circumcenters, Euler lines, and Feuerbach spheres, using duality techniques.

Findings

Generalized properties of simplices in normed spaces.

New theorems on angular bisectors and in-/exspheres.

Connections between Euclidean and non-Euclidean simplex geometry.

Abstract

There are many problems and configurations in Euclidean geometry that were never extended to the framework of (normed or) finite dimensional real Banach spaces, although their original versions are inspiring for this type of generalization, and the analogous definitions for normed spaces represent a promising topic. An example is the geometry of simplices in non-Euclidean normed spaces. We present new generalizations of well known properties of Euclidean simplices. These results refer to analogues of circumcenters, Euler lines, and Feuerbach spheres of simplices in normed spaces. Using duality, we also get natural theorems on angular bisectors as well as in- and exspheres of (dual) simplices.