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MINKOWSKI PRODUCT OF CONVEX SETS AND PRODUCT NUMERICAL RANGE
Li, Chi-Kwong Pelejo, Diane Christine Poon, Yiu-Tung Wang, Kuo-Zhong
Li, Chi-Kwong
Pelejo, Diane Christine
Poon, Yiu-Tung
Wang, Kuo-Zhong
Abstract
Let K-1, K-2 be two compact convex sets in C. Their Minkowski product is the set K1K2 = {ab : a is an element of K-1, b is an element of K-2}. We show that the set K1K2 is star-haped 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 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.
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2016-11-04
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oam_10_53_abs.pdf
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Mathematics
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10.7153/oam-10-53