In this paper we introduce and investigate similarity measures for convex polyhedra based on Minkowski addition and inequalities for the mixed volume, volume and surface area related to the Brunn-Minkowski theory. All measures considered are invariant under translations; furthermore, they may also be invariant under subgroups of the aane transformation group. For the case of rotation and scale ...

Theorem 7.1 (Brunn-Minkowski). If A, B ⊆ R n satisfy some mild assumptions (in particular, convexity suffices), then [vol (A + B)] 1 n ≥ [vol (A)] 1 n + [vol (B)] 1 n where A + B = { a + b : a ∈ A and b ∈ B}.

Parts of the Brunn-Minkowski theory can be extended to hedgehogs, which are envelopes of families of affine hyperplanes parametrized by their Gauss map. In [F], F. Fillastre introduced Fuchsian convex bodies, which are the closed convex sets of Lorentz-Minkowski space that are globally invariant under the action of a Fuchsian group. In this paper, we undertake a study of plane Lorentzian and Fu...

One of the famous mathematical inequalities is Minkowski’s inequality. It is an important inequality from both mathematical and application points of view. In this paper, a Minkowski type inequality for fuzzy integrals is studied. The established results are based on the classical Minkowski inequality for integrals. Moreover, a generalized Minkowski type inequality for fuzzy integrals is sugges...