Isometric embeddings of polyhedra into Euclidean space

Geometric Quasi - Isometric Embeddings Into

We show that F has an infinite family of quasi-isometrically embedded subgroups of the form F m × Zn, for integral m, n ≥ 0. These subgroups have simple geometric but more complicated algebraic descriptions We present them to illustrate the intricate geometry of Thompson’s group F as well as the interplay between its standard finite and infinite presentations.

Rigid Ball-Polyhedra in Euclidean 3-Space

A ball-polyhedron is the intersection with non-empty interior of finitely many (closed) unit balls in Euclidean 3-space. One can represent the boundary of a ballpolyhedron as the union of vertices, edges, and faces defined in a rather natural way. A ball-polyhedron is called a simple ball-polyhedron if at every vertex exactly three edges meet. Moreover, a ball-polyhedron is called a standard ba...

Embeddings of Surfaces in Euclidean Three-space

Rigidity of ball-polyhedra in Euclidean 3-space

In this paper we introduce ball-polyhedra as finite intersections of congruent balls in Euclidean 3-space. We define their duals and study their face-lattices. Our main result is an analogue of Cauchy’s rigidity theorem. © 2004 Elsevier Ltd. All rights reserved.

Mapping preferences into Euclidean space

Understanding and modeling human preferences is one of the key problems in applications ranging from marketing to automated recommendation. In this paper, we focus on learning and analyzing the preferences of consumers regarding food products. In particular, we explore machine learning methods that embed consumers and products in an Euclidean space such that their relationship to each other mod...