The Brunn–Minkowski Theorem asserts that μd(A+B)1/d ≥ μd(A) + μd(B) 1/d for convex bodies A,B ⊆ R , where μd denotes the d-dimensional Lebesgue measure. It is well known that equality holds if and only if A and B are homothetic, but few characterizations of equality in other related bounds are known. Let H be a hyperplane. Bonnesen later strengthened this bound by showing μd(A+B)≥ (M1/(d−1) +N1...

Random closed sets (RACS) in the d–dimensional Euclidean space are considered, whose realizations belong to the extended convex ring. A family of nonparametric estimators is investigated for the simultaneous estimation of the vector of all specific Minkowski functionals (or, equivalently, the specific intrinsic volumes) of stationary RACS. The construction of these estimators is based on a repr...

For a general family of graphs on Zn, we translate the edge-isoperimetric problem into a continuous isoperimetric problem in Rn. We then solve the continuous isoperimetric problem using the Brunn-Minkowski inequality and Minkowski’s theorem on mixed volumes. This translation allows us to conclude, under a reasonable assumption about the discrete problem, that the shapes of the optimal sets in t...

Centroid and difference bodies define SL(n) equivariant operators on convex bodies and these operators are valuations with respect to Minkowski addition. We derive a classification of SL(n) equivariant Minkowski valuations and give a characterization of these operators. We also derive a classification of SL(n) contravariant Minkowski valuations and of Lp-Minkowski valuations. 2000 AMS subject c...

This paper is devoted to similarity and symmetry measures for convex shapes whose deenition is based on Minkowski addition and the Brunn-Minkowski inequality. This means in particular that these measures are region-based, in contrast to most of the literature, where one considers contour-based measures. All measures considered in this paper are invariant under translations; furthermore, they ca...

for any compact sets K, T ⊂ R, where (K +T )/2 = {(x+ y)/2; x ∈ K, y ∈ T} is half of the Minkowski sum of K and T , and where V oln stands for the Lebesgue measure in R. Equality in (1) holds if and only if K is a translate of T and both are convex, up to a set of measure zero. The literature contains various stability estimates for the Brunn-Minkowski inequality, which imply that when there is...