Theorem 19.7.566339C0E3C21

Proof. Let $K_j\to K$ in Hausdorff distance. All bodies lie in a fixed ball. Every point outside $K$ eventually lies outside $K_j$, because its distance from $K$ is positive. If $K$ has interior, every point $x$ of its interior eventually lies in $K_j$: choose $\rho>0$ with $x+\rho\overline B\subset K$; uniform convergence of support functions then gives $\langle u,x\rangle<h_{K_j}(u)$ for every unit $u$, which by separation implies $x\in K_j$.

The boundary of a convex body has Lebesgue measure zero, as proved in the relative-boundary lemma. Thus the indicator functions converge almost everywhere, and dominated convergence on the fixed ball proves volume convergence. If $K$ has empty interior, it lies in a proper affine subspace and has measure zero; the outside-point argument alone again gives indicator convergence almost everywhere. In dimension zero the normalized volume is always $1$.