A note on some generalizations of Monge's theorem

TL;DR

This paper extends Monge's theorem to higher dimensions, convex bodies, and different geometries, providing new generalizations and analogs in Euclidean, spherical, and hyperbolic spaces.

Contribution

It introduces generalized versions of Monge's theorem for homothetic sets, independent points, and non-Euclidean geometries, broadening its applicability.

Findings

Generalization to $n+1$ homothetic sets in $E^n$

Version for $n+1$ independent points in $E^n$

Analog of Monge's theorem in spherical and hyperbolic geometries

Abstract

We generalize Monge's theorem for $n+1$ pairwise homothetic sets (in particular convex bodies) in $E^n$ in place of three disks in $E^2$. We also present a version for $n+1$ independent points of $E^n$. It also includes the reverse statement. Moreover, we give an analogon of Monge's theorem for the $n$-dimensional sphere and hyperboloid model of the hyperbolic space.