Simultaneous inner and outer approximation of shapes (Q1201744)

A new metric is introduced on the set of compact Euclidean bodies: for two of them \(L(P,Q)\) is the smallest ratio \(r/s\) of scaling factors \(r>0\), \(s>0\) where \(sQ'\subseteq R\subseteq rQ''\) and \(Q'\), \(Q''\) are translates of \(Q\). This is motivated by design of heuristic algorithms for motion planning and object placement problems. Estimates and efficient algorithms are given for the cases where \(P\) is a convex polygon and \(Q\) is a triangle. Upper bounds for other \(k\)-gons are also considered.