Minkowski roots of complex sets

Abstract

An n-th Minkowski root /spl otimes//sup 1/n/ A of a given complex set A is defined by the property {z/sub 1/z/sub 2/...z/sub n/|z/sub i//spl isin//spl otimes//sup 1/n/ A}/spl equiv/A, i.e., the set of all products of n independently chosen values from /spl otimes//sup 1/n/ is identical to A. Hence, the n-th Minkowski power of /spl otimes//sup 1/n/A yields the original set A. Minkowski root extractions are fundamental operations in the Minkowski geometric algebra of complex sets: depending on the nature of A, subtle issues concerning the existence, uniqueness and minimality or maximality of /spl otimes//sup 1/n/A may arise. For a domain A with a smooth boundary that is strictly logarithmically convex, we show that each connected component of the "ordinary" root A/sup 1/n/={z|z/sup n//spl isin/A} is a Minkowski n-th root. /spl otimes//sup 1/n/A has a more intricate structure, however when /spl part/A has logarithmic inflections. For example, if A is a circular disk, /spl otimes//sup 1/n/A is (a single loop of) the "n-th order" ovals of Cassini or lemniscate of Bernoulli when 0/spl notin/A or 0/spl isin//spl part/A, respectively. But when 0 is in the interior of A, a composite curve (portions of the Cassini ovals and a higher-order curve) is required to describe /spl otimes//sup 1/n/A.