M-addition

M  -addition, denoted by ⊕M⊕M, is a way of combining sets in a vector space which generalizes Minkowski addition +. Under certain circumstances Minkowski addition satisfies (K∩L)+C=(K+C)∩(L+C)(K∩L)+C=(K+C)∩(L+C) and (K∪L)+(K∩L)=K+L(K∪L)+(K∩L)=K+L. We prove parallel properties for M-addition and classify all compact sets M   for which (K∪L)⊕M(K∩L)=K⊕ML(K∪L)⊕M(K∩L)=K⊕ML. Minkowski addition also fulfills conv(∑j=1mAj)=∑j=1mconvAj, and we classify all sets M   for which conv(⊕M(A1,…,Am))=⊕convM(convA1,…,convAm), the natural M  -addition generalization. Corollaries drawn from this result include conditions for when ⊕M⊕M maps convex polytopes to convex polytopes and an extension of the Shapley–Folkman lemma to M-addition.