Minkowski product of convex sets and product numerical range

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

Research output: Contribution to journalArticle

2 Scopus citations

Abstract

Let K1,K2 be two compact convex sets in C. Their Minkowski product is the set K1K2 = {ab: a ∈ K1,b ∈ K2}. We show that the set K1K2 is star-shaped if K1 is a line segment or a circular disk. Examples for K1 and K2 are given so that K1 and K2 are triangles (including interior) and K1K2 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.

Original languageEnglish
Article numberoam-10-53
Pages (from-to)945-965
Number of pages21
JournalOperators and Matrices
Volume10
Issue number4
DOIs
StatePublished - 1 Dec 2016

Keywords

  • Convex sets
  • Minkowski product
  • Numerical range

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