We prove that for any norm · in the d-dimensional real vector space V and for any odd n > 0 there is a non-negative polynomial p(x), x ∈ V of degree 2n such that p 1 2n (x) ≤ x ≤ n + d − 1 n 1 2n p 1 2n (x). Corollaries and polynomial approximations of the Minkowski functional of a convex body are discussed.

The no"t polygon is a powerful and e!ective tool for handling the geometry required for a range of solution approaches to two-dimensional irregular cutting-stock problems. However, unless all the pieces are convex, it is widely perceived as being di$cult to implement, and its use has therefore been somewhat limited. The primary purpose of this paper is to correct this misconception by introduci...

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...

In this note we reconsider linearised metric perturbations in the one-brane Randall-Sundrum Model. We present a simple formalism to describe metric perturbations caused by matter perturbations on the brane and remedy some misconceptions concerning the constraints imposed on the metric and matter perturbations by the presence of the brane. An interesting alternative to standard Kaluza-Klein comp...

We prove that for any norm k k in the d-dimensional real vector space V and for any odd n > 0 there is a non-negative polynomial p(x), x 2 V of degree 2n such that p 1 2n (x) kxk n + d ? 1 n 1 2n p 1 2n (x): Corollaries and polynomial approximations of the Minkowski functional of a convex body are discussed.

There is much discussion of scenarios where the space-time coordinates x are noncommutative. The discussion has been extended to include nontrivial anticommutation relations among spinor coordinates in superspace. A number of authors have studied field theoretical consequences of the deformation of N = 1 superspace arising from nonanticommutativity of coordinates θ, while leaving θ̄’s anticommut...

In this paper we propose a method to characterize and estimate the variations of a random convex set Ξ0 in terms of shape, size and direction. The mean n-variogram γ Ξ0 : (u1 · · ·un) 7→ E[νd(Ξ0∩(Ξ0−u1) · · ·∩ (Ξ0 − un))] of a random convex set Ξ0 on R reveals information on the n order structure of Ξ0. Especially we will show that considering the mean n-variograms of the dilated random sets Ξ0...

Abstract The main constraint on relative position of geometric objects, used in spatial planning for computing the C-space maps (for example, in robotics, CAD, and packaging), is the relative non-overlapping of objects. This is the simplest constraint in which the minimum translational distance between objects is greater than zero, or more generally, than some positive value. We present a techn...

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 define complex Minkowski superspace in 4 dimensions as the big cell inside a complex flag supermanifold. The complex conformal supergroup acts naturally on this super flag, allowing us to interpret it as the conformal compactification of complex Minkowski superspace. We then consider real Minkowski superspace as a suitable real form of the complex version. Our methods are group theoretic, ba...