Minkowski product of convex sets and product numerical range
نویسندگان
چکیده
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.
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