We investigate how the Minkowski sum of two polytopes affects their graph and, in particular, their diameter. We show that the diameter of the Minkowski sum is bounded below by the diameter of each summand and above by, roughly, the product between the diameter of one summand and the number of vertices of the other. We also prove that both bounds are sharp. In addition, we obtain a result on polytope decomposability. More precisely, given two polytopes $P$ and $Q$, we show that $P$ can be written as a Minkowski sum with a summand homothetic to $Q$ if and only if $P$ has the same number of vertices as its Minkowski sum with $Q$.

The Gauss−Minkowskicorrespondence in ℝ2 states the existence of a homeomorphism between theprobability measures μ on [0,2π] such that \hbox{$\int_0^{2\pi} {\rm e}^{ix}{\rm d}\mu(x)=0$}${{}^{\mathrm{\int }}}_{\mathrm{0}}^{\mathrm{2}\mathit{\pi }}{\mathrm{e}}^{\mathit{ix}}\mathrm{d}\mathit{\mu }\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\mathrm{0}$ and the compact convex sets (CCS) of the plane withperimeter 1. In this article, we bring out explicit formulas relating the border of a CCSto its probability measure. As a consequence, we show that some natural operations on CCS– for example, the Minkowski sum – have natural translations in terms of probabilitymeasure operations, and reciprocally, the convolution of measures translates into a newnotion of convolution of CCS. Additionally, we give a proof that a polygonal curveassociated with a sample of n random variables (satisfying \hbox{$\int_0^{2\pi} {\rm e}^{ix}{\rm d}\mu(x)=0$}${{}^{\mathrm{\int }}}_{\mathrm{0}}^{\mathrm{2}\mathit{\pi }}{\mathrm{e}}^{\mathit{ix}}\mathrm{d}\mathit{\mu }\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\mathrm{0}$) converges to a CCS associated with μ at speed √n, a result much similar to the convergence of theempirical process in statistics. Finally, we employ this correspondence to present modelsof smooth random CCS and simulations.

We prove large deviation results for Minkowski sums Sn of independent and identically distributed random compact sets where we assume that the summands have a regularly varying distribution and finite expectation. The main focus is on random convex compact sets. The results confirm the heavy-tailed large deviation heuristics: ‘large’ values of the sum are essentially due to the ‘largest’ summand. These results extend those in Mikosch, Pawlas and Samorodnitsky (2011) for generally nonconvex sets, where we assumed that the normalization of Sn grows faster than n.

Motivated by the well-posedness of birth-and-growth processes, a stochastic geometric differential equation and, hence, a stochastic geometric dynamical system are proposed. In fact, a birth-and-growth process can be rigorously modeled as a suitable combination, involving the Minkowski sum and the Aumann integral, of two very general set-valued processes representing nucleation and growth dynamics, respectively. The simplicity of the proposed geometric approach allows to avoid problems of boundary regularities arising from an analytical definition of the front growth. In this framework, growth is generally anisotropic and, according to a mesoscale point of view, is non local, i.e. at a fixed time instant, growth is the same at each point of the space.