XIV.—Notes on Inscribed Strigils

The Pentagram Integrals on Inscribed Polygons

The pentagram map is a completely integrable system defined on the moduli space of polygons. The integrals for the system are certain weighted homogeneous polynomials, which come in pairs: E1, O2, E2, O2, . . . In this paper we prove that Ek = Ok for all k, when these integrals are restricted to the space of polygons which are inscribed in a conic section. Our proof is essentially a combinatori...

Containment and Inscribed Simplices

Let K and L be compact convex sets in Rn. The following two statements are shown to be equivalent: (i) For every polytope Q ⊆ K having at most n+ 1 vertices, L contains a translate of Q. (ii) L contains a translate of K. Let 1 ≤ d ≤ n − 1. It is also shown that the following two statements are equivalent: (i) For every polytope Q ⊆ K having at most d+ 1 vertices, L contains a translate of Q. (i...

Enumeration of inscribed polyominos

We introduce a new family of polyominos that are inscribed in a rectangle of given size for which we establish a number of exact formulas and generating functions. In particular, we study polyominos inscribed in a rectangle with minimum area and minimum area plus one. These results are then used for the enumeration of lattice trees inscribed in a rectangle with minimum area plus one. Résumé. No...

A New Palmprint Segmentation Method Based on an Inscribed Circle

Neighborly inscribed polytopes and Delaunay triangulations

We prove that there are superexponentially many combinatorially distinct d-dimensional neighborly Delaunay triangulations on n points. These are the first examples of neighborly Delaunay triangulations that cannot be obtained via a stereographic projection of an inscribed cyclic polytope, and provide the current best lower bound for the number of combinatorial types of Delaunay triangulations. ...