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.
- Pub Date:
- January 2013
- Mathematics - Metric Geometry;
- Mathematics - Differential Geometry;
- Mathematics - Functional Analysis;
- 52C15 37J45
- Final submitted manuscript. Geometriae Dedicata, 2013