Minkowski product of convex sets and product numerical range

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

Minkowski product of convex sets and product numerical range

Metric Geometry 2015-05-21 v1

Authors: Chi-Kwong Li , Diane Christine Pelejo , Yiu-Tung Poon , Kuo-Zhong Wang

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.

Keywords

Minkowski product convex sets star-shaped set numerical range quantum information product numerical range counterexample

Cite

@article{arxiv.1505.05382,
  title  = {Minkowski product of convex sets and product numerical range},
  author = {Chi-Kwong Li and Diane Christine Pelejo and Yiu-Tung Poon and Kuo-Zhong Wang},
  journal= {arXiv preprint arXiv:1505.05382},
  year   = {2015}
}

Minkowski product of convex sets and product numerical range

Abstract

Keywords

Cite

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