Minkowski addition

The Minkowski sum of two sets $A$, $B$ in $n$-dimensional Euclidean space $\mathbf{E} ^ { n }$ is defined as the set

\begin{equation*} A + B : = \{ a + b : a \in A , b \in B \}; \end{equation*}

one also defines $\lambda A : = \{ \lambda a : a \in A \}$ for real $\lambda > 0$. Coupled with the notion of volume, this Minkowski addition leads to the Brunn–Minkowski theorem and is the basis for the Brunn–Minkowski theory of convex bodies (i.e., compact convex sets).

Repeated Minkowski addition of compact sets has a convexifying effect; this is made precise by the Shapley–Folkman–Starr theorem.

The structure of Minkowski addition is well studied on the space $\mathcal{K} ^ { n }$ of convex bodies in $\mathbf{E} ^ { n }$. $\mathcal{K} ^ { n }$ with Minkowski addition and multiplication by non-negative scalars is a convex cone. The mapping $K \mapsto h _ { K }$, where

\begin{equation*} h _ { K } ( u ) : = \operatorname { max } \{ \langle x , u \rangle : x \in K \}, \end{equation*}

\begin{equation*} u \in S ^ { n - 1 } : = \{ v \in {\bf E} : \langle v , v \rangle = 1 \} \end{equation*}

($\langle \, .\, ,\, . \, \rangle$ being the scalar product) is the support function, maps this cone isomorphically into the space $C ( S ^ { n - 1 } )$ of continuous real functions on $S ^ { n - 1 }$. The image is precisely the closed convex cone of restrictions of sublinear functions. (For corresponding results in topological vector spaces, see [a2] and its bibliography.)

For convex bodies $K , L \in {\cal K} ^ { n }$, the body $L$ is called a summand of $K$ if there exists an $M \in \mathcal{K} ^ { n }$ such that $K = L + M$. Each summand of $K$ is a non-empty intersection of a family of translates of $K$; the converse is true for $n = 2$. $K$ is called indecomposable if every summand of K is of the form $\lambda K + t$ with $\lambda > 0$ and $t \in \mathbf{E} ^ { n }$. In the plane $\mathbf{E} ^ { 2 }$, the indecomposable convex bodies are precisely the segments and the triangles. For $n \geq 3$, every simplicial convex polytope in $\mathbf{E} ^ { n }$ is indecomposable, hence most convex bodies (in the Baire category sense, cf. also Baire set) are indecomposable.

A mapping $\phi$ from $\mathcal{K} ^ { n }$ into an Abelian group is called Minkowski additive if $\phi ( K + L ) = \phi ( K ) + \phi ( L )$ for all $K , L \in {\cal K} ^ { n }$. Such mappings are special valuations and play a particular role in the investigation of valuations on convex bodies. Common examples are the mean width and the Steiner point $s$. A surjective mapping $\psi : \mathcal{K} ^ { n } \rightarrow \mathcal{K} ^ { n }$ with $\psi ( K + L ) = \psi ( K ) + \psi ( L )$ that commutes with rigid motions and is continuous with respect to the Hausdorff metric is trivial, namely of the form $\psi ( K ) = \lambda [ K - s ( K ) ] + s ( K )$ with $\lambda \neq 0$.

References

[a1]	R. Schneider, "Convex bodies: the Brunn–Minkowski theory" , Cambridge Univ. Press (1993)
[a2]	R. Urbanski, "A generalization of the Minkowski–Rådström–Hörmander theorem" Bull. Acad. Polon. Sci. Ser. Sci. Math., Astr., Phys. , 24 (1976) pp. 709 – 715