Minkowski product of convex sets and product numerical range

TL;DR

This paper investigates the Minkowski product of convex sets, showing star-shapedness under certain conditions, providing counterexamples to a conjecture in quantum information, and discussing related open problems.

Contribution

It establishes conditions for star-shapedness of Minkowski products and provides counterexamples to a conjecture on product numerical range.

Findings

K_1K_2 is star-shaped if K_1 is a line segment or disk

Counterexamples show K_1K_2 need not be star-shaped

Negative answer to a conjecture in quantum information science

Abstract

Let $K_1, K_2$ be two compact convex sets in $\mathit{C}$. Their Minkowski product is the set $K_1K_2 = \{ab: a \in K_1, b\in K_2\}$. We show that the set $K_1K_2$ 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 $K_1K_2$ 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.