Minkowski product of convex sets and product numerical range

Chi-Kwong Li; Diane Christine Pelejo; Yiu-Tung Poon; Kuo-Zhong Wang

paper

Minkowski product of convex sets and product numerical range

arXiv:1505.05382

Abstract

Let $K_{1}, K_{2}$ be two compact convex sets in $C$ . Their Minkowski product is the set $K_{1} K_{2} = {ab : a \in K_{1}, b \in K_{2}}$ . We show that the set $K_{1} K_{2}$ is star-shaped if $K_{1}$ is a line segment or a circular disk. Examples for $K_{1}$ and $K_{2}$ are given so that $K_{1}$ and $K_{2}$ are triangles (including interior) and $K_{1} K_{2}$ is not star-shaped. This gives a negative answer to a conjecture by Puchala et. al concerning the product numerical range in the study of quantum information science. Additional results and open problems are presented.