Minkowski complexes and convex threshold dimension

作者：Frick Florian^*; Sanyal Raman

来源：Journal of Combinatorial Theory - Series A, 2017, 151: 202-206.

DOI：10.1016/j.jcta.2017.04.010

• Minkowski sum
• Minkowski complexes
• Threshold complexes
• Convex threshold dimension
• Discrete mixed volume

For a collection of convex bodies P-1,...,P-n subset of R-d containing the origin, a Minkowski complex is given by those subsets whose Minkowski sum does not contain a fixed basepoint. Every simplicial complex can be realized as a Minkowski complex and for convex bodies on the real line, this recovers the class of threshold complexes. The purpose of this note is the study of the convex threshold dimension of a complex, that is, the smallest dimension in which it can be realized as a Minkowski complex. In particular, we show that the convex threshold dimension can be arbitrarily large. This is related to work of Chvatal and Hammer (1977) regarding forbidden subgraphs of threshold graphs. We also show that convexity is crucial this context.

• 出版日期2017-10