Inextensible domains

TL;DR

This paper characterizes inextensible convex domains with respect to lattice covering, linking them to outer billiard triangles, and addresses a related conjecture by Genin and Tabachnikov.

Contribution

It provides a complete characterization of symmetric inextensible domains and connects the concept to outer billiard triangles, also addressing an existing conjecture.

Findings

Origin-symmetric inextensible domains have a circle of outer billiard triangles.

Addresses a conjecture by Genin and Tabachnikov on convex domains with outer billiard triangles.

Shows that such domains are characterized by a specific geometric property.

Abstract

We develop a theory of planar, origin-symmetric, convex domains that are inextensible with respect to lattice covering, that is, domains such that augmenting them in any way allows fewer domains to cover the same area. We show that origin-symmetric inextensible domains are exactly the origin-symmetric convex domains with a circle of outer billiard triangles. We address a conjecture by Genin and Tabachnikov about convex domains, not necessarily symmetric, with a circle of outer billiard triangles, and show that it follows immediately from a result of Sas.