Decomposition of closed convex sets (Q1113085)

\textit{W. Weil} [Arch. Math. 34, 283-288 (1980; Zbl 0424.52002)] proved that a convex body L is a summand of a convex body K iff for each translate L'\(\not\subset K\) of L, the set of all points of L' farthest from K is a support set of L'. The author generalizes this result to the family \(C^ d\) of (not necessarily bounded) non-empty convex sets in \({\mathbb{R}}^ d\). He defines \(L\in C^ d\) to be a pseudo-summand of \(K\in C^ d\) if K is the closure of \(L+M\) for some \(M\in C^ d\). Then the following theorem is proved: \(L\in C^ d\) is a pseudosummand of \(K\in C^ d\) iff for each translate L' of L there is exactly one point in D(L',K) farthest from the origin. Here D(L',K) denotes the closure of the set of vectors \(x-p_ K(x)\), \(x\in L\), where \(p_ K\) denotes the nearest-point-map with respect to K.