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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 $\lambda &gt; 0$. Coupled with the notion of [[Volume|volume]], this Minkowski addition leads to the [[Brunn–Minkowski theorem|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 [[#References|[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 $\lambda &gt; 0$ 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|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|Hausdorff metric]] is trivial, namely of the form $\psi ( K ) = \lambda [ K - s ( K ) ] + s ( K )$ with $\lambda \neq 0$.

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  R. Schneider,   "Convex bodies: the Brunn–Minkowski theory" , Cambridge Univ. Press  (1993)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  R. Urbanski,   "A generalization of the Minkowski–Rådström–Hörmander theorem"  ''Bull. Acad. Polon. Sci. Ser. Sci. Math., Astr., Phys.'' , '''24'''  (1976)  pp. 709 – 715</td></tr></table>