This paper deals with the containment problem under homothetics which has the minimal enclosing ball (MEB) problem as a prominent representative. We connect the problem to results in classic convex geometry and introduce a new series of radii, which we call core-radii. For the MEB problem, these radii have already been considered from a different point of view and sharp inequalities between them are known. In this paper sharp inequalities between core-radii for general containment under homothetics are obtained.Moreover, the presented inequalities are used to derive sharp upper bounds on the size of core-sets for containment under homothetics. In the MEB case, this yields a tight (dimension-independent) bound for the size of such core-sets. In the general case, we show that there are core-sets of size linear in the dimension and that this bound stays sharp even if the container is required to be symmetric.
Abstract. This paper deals with the three types of regular polytopes which exist in all dimensions -regular simplices, cubes and regular cross-polytopesand their outer and inner radii. While the inner radii of regular simplices are well studied, only a few cases are solved for the outer radii. We give a lower bound on these radii, and show that this bound is tight in almost 3 out of 4 dimensions. In a further section we complete the results about inner and outer radii of general boxes and cross-polytopes. Finally, because cubes and regular cross-polytopes are radii-minimal projections of simplices, we show that it is possible to deduce the results about their radii from the results about the outer radii of simplices. Introduction and basic notationsThere are three classes of regular polytopes which last in general d-space: regular simplices, (hyper-) cubes, and regular cross-polytopes. In this paper we investigate the inner and outer j-radii of this polytopes. Here the inner j-radius of a body is defined as the radius of a biggest j-ball fitting into the body, and the outer j-radius as the smallest radius of a circumball of an orthogonal projection of the body onto any j-space.While the inner radii of regular simplices are well studied [1], very less is known about their outer radii. We give a lower bound on these radii, and show that this bound is tight in almost 3 out of 4 cases. An important step towards this result is the investigation of quasi isotropic polytopes (Kawashima called them π-polytopes [9], but we prefer to call them isotropic as they are in an isotropic position in the sense of [5]). Specifically, we will show that the existence of a quasi-isotropic j-dimensional polytope with d + 1 vertices is equivalent to the existence of a projection of the regular simplex such that the lower bound is attained.In a further section we investigate the radii of general boxes and cross-polytopes. While the inner radii of boxes were computed in [4], nothing could be found about their outer radii in the literature. We close this gap, as well as we transfer the Date: October 31, 2018.
Many classical geometric inequalities on functionals of convex bodies depend on the dimension of the ambient space. We show that this dimension dependence may often be replaced (totally or partially) by different symmetry measures of the convex body. Since these coefficients are bounded by the dimension but possibly smaller, our inequalities sharpen the original ones. Since they can often be computed efficiently, the improved bounds may also be used to obtain better bounds in approximation algorithms.
Minimal containment problems arise in a variety of applications, such as shape fitting and packing problems, data clustering, pattern recognition, or medical surgery. Typical examples are the smallest enclosing ball, cylinder, slab, box, or ellipsoid of a given set of points.Here we focus on one of the most basic problems: minimal containment under homothetics, i.e., covering a point set by a minimally scaled translation of a given container. Besides direct applications this problem is often the base in solving much harder containment problems and therefore fast solution methods are needed, especially in moderate dimensions. While in theory the ellipsoid method suffices to show polynomiality in many cases, extensive studies of implementations exist only for Euclidean containers. Indeed, many applications require more complicated containers.In Plastria (Eur. J. Oper. Res. 29:98-110, 1987) the problem is discussed in a more general setting from the facility location viewpoint and a cutting plane method is suggested. In contrast to Plastria (Eur. J. Oper. Res. 29:98-110, 1987), our approach relies on more and more accurate approximations of the container. For facet and vertex presented polytopal containers the problem can be formulated as an LP, and for many general containers as an SOCP. The experimental section of the paper compares those formulations to the cutting plane method, showing that it outperforms the LP formulations for vertex presented containers and the SOCP formulation for some problem instances. L. Roth has been supported by the "Deutsche Forschungsgemeinschaft" through the graduate program "Angewandte
Abstract. We provide an algebraic framework to compute smallest enclosing and smallest circumscribing cylinders of simplices in Euclidean space E n . Explicitly, the computation of a smallest enclosing cylinder in E 3 is reduced to the computation of a smallest circumscribing cylinder. We improve existing polynomial formulations to compute the locally extreme circumscribing cylinders in E 3 and exhibit subclasses of simplices where the algebraic degrees can be further reduced. Moreover, we generalize these efficient formulations to the n-dimensional case and provide bounds on the number of local extrema. Using elementary invariant theory, we prove structural results on the direction vectors of any locally extreme circumscribing cylinder for regular simplices.
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