The Brunn-Minkowski inequality asserts the concavity of \(V^{1/d}(X)\) under vector addition of measurable sets S in the Euclidean space \({\mathbb{R}}^ d\), where V signifies Lebesgue measure. Let \({\mathcal K}\) be the set of non-empty, compact sets in \({\mathbb{R}}^ d\), metrized by the Hausdorff metric; define the norm \(\| K\|\) of an element of \({\mathcal K}\) to be \(\max \{\| x\| :\) \(x\in K\}\). A random set X is a Borel measurable map \(X: \Omega \to {\mathcal K}\) for some probability space (\(\Omega\),\({\mathcal A},P)\). A random vector \(\alpha: \Omega \to {\mathbb{R}}^ d\) is a selection of X if \(P[\alpha \in X]=1\). The author defines the expectation E X of X to be \[ \{E_{\alpha}:\;\alpha \text{ is a selection of X and } E \| \alpha \| <\infty \}; \] here the author uses an adaptation of the R. J. Aumann integral [J. Math. Analysis Appl. 12, 1-12 (1965; Zbl 0163.063)]. The condition \(E\| X\| <\infty\) guarantees that \(E X\in {\mathcal K}.\)
The main result is a generalization of the Brunn-Minkowski theorem: \(V^{1/d}(E X)\geq E V^{1/d}(X)\), when the random set X satisfies \(E \| X\| <\infty\). This is applied to prove an inequality of G. S. Mudholkar [Proc. Amer. Math. Soc. 17, 1327-1333 (1966; Zbl 0163.069)]:
if \(\Gamma\) is a group of linear, measure-preserving maps of \({\mathbb{R}}^ d\) onto \({\mathbb{R}}^ d\), E is a convex, \(\Gamma\)-invariant region of \({\mathbb{R}}^ d\), and f is a non-negative, \(\Gamma\)-invariant function of \({\mathbb{R}}^ d\) whose level sets are convex, then the integral over E of \(f(x+z)\) is not less than the integral over E of \(f(x+y)\), both taken with respect to x, where z is any point of the convex hull of the \(\Gamma\)-orbit of y.