A description of continuous rigid motion compatible Minkowski valuations is established. As an application we present a Brunn–Minkowski type inequality for intrinsic volumes of these valuations.

We present an incremental algorithm to estimate the penetration depth between convex polytopes in 3D. The algorithm incrementally seeks a “locally optimal solution” by walking on the surface of the Minkowski sums. The surface of the Minkowski sums is computed implicitly by constructing a local Gauss map. In practice, the algorithm works well when there is high motion coherence in the environmen...

The classical Brunn-Minkowski theory for convex bodies was developed from a few basic concepts: support functions, Minkowski combinations, and mixed volumes. As a special case of mixed volumes, the Quermassintegrals are important geometrical quantities of a convex body, and surface area measures are local versions of Quermassintegrals. The Christoffel-Minkowski problem concerns with the existen...

In this paper we study translation surfaces with the non-degenerate third fundamental form in Lorentz- Minkowski space $mathbb{L}^{3}$. As a result, we classify translation surfaces satisfying an equation in terms of the position vector field and the Laplace operator with respect to the third fundamental form $III$ on the surface.

Recently, Eisenbrand, Pach, Rothvoß, and Sopher studied the function M(m, n), which is the largest cardinality of a convexly independent subset of the Minkowski sum of some planar point sets P and Q with |P | = m and |Q| = n. They proved that M(m, n) = O(m2/3n2/3+m+n), and asked whether a superlinear lower bound exists for M(n, n). In this note, we show that their upper bound is the best possib...

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...

Computing the Minkowski sum of two polyhedra exactly has been shown difficult. Despite its fundamental role in many geometric problems in robotics, to the best of our knowledge, no 3-d Minkowski sum software for general polyhedra is available to the public. One of the main reasons is the difficulty of implementing the existing methods. There are two main approaches for computing Minkowski sums:...

We investigate the hypothesis that the macroscopic properties of a porous material can be determined from limited morphological information. Specifically, we investigate this hypothesis for the Minkowski functionals of two-phase media in 2-D. We look at two methods for generating samples with desired Minkowski functionals: the Gibbs sampler and the Metropolis-Hastings algorithm. The Metropolis-...

Minkowski valuations provide a systematic framework for quantifying different aspects of morphology. In this paper we apply vector- and tensor-valued Minkowski valuations to neuronal cells from the cat's retina in order to describe their morphological structure in a comprehensive way. We introduce the framework of Minkowski valuations, discuss their implementation for neuronal cells and show ho...

This paper extends some results for the weighted Moore–Penrose inverse A+M,N in Hilbert space to the so-called weighted Minkowski inverse A⊕M,N of an arbitrary rectangular matrix A ∈ Mm,n in Minkowski spaces μ. Four methods are also used for approximating the weighted Minkowski Inverse A⊕M,N . These methods are: Borel summable, Euler–Knopp summable, Newton–Raphson and Tikhonov’s methods. c © 20...