# Self-Polar Polytopes

## TL;DR

This paper explores self-polar polytopes, a special class of convex polytopes related to their polar sets, focusing on their existence, structure, and applications in graph theory.

## Contribution

It provides new insights into the existence, construction, and facial structure of self-polar polytopes, and discusses their role within self-dual polytopes.

## Key findings

- Self-polar polytopes can be constructed for various dimensions.
- They have applications in graph coloring and constructing graphs with high chromatic number.
- The paper clarifies the relationship between self-polar and self-dual polytopes.

## Abstract

Self-polar polytopes are convex polytopes that are equal to an orthogonal transformation of their polar sets. These polytopes were first studied by Lov\'{a}sz as a means of establishing the chromatic number of distance graphs on spheres, and they can also be used to construct triangle-free graphs with arbitrarily high chromatic number. We investigate the existence, construction, facial structure, and practical applications of self-polar polytopes, as well as the place of these polytopes within the broader set of self-dual polytopes.