منابع مشابه
Geometric Construction of Coordinates for Convex Polyhedra using Polar Duals
A fundamental problem in geometry processing is that of expressing a point inside a convex polyhedron as a combination of the vertices of the polyhedron. Instances of this problem arise often in mesh parameterization and 3D deformation. A related problem is to express a vector lying in a convex cone as a non-negative combination of edge rays of this cone. This problem also arises in many applic...
متن کاملShadow systems and volumes of polar convex bodies
We prove that the reciprocal of the volume of the polar bodies, about the Santaló point, of a shadow system of convex bodies Kt, is a convex function of t. Thus extending to the non-symmetric case a result of Campi and Gronchi. The case that the reciprocal of the volume is an affine function of t is also investigated and is characterized under certain conditions. We apply these results to prove...
متن کاملShadows of Convex Bodies
It is proved that if C is a convex body in R" then C has an affine image C (of nonzero volume) so that if P is any 1-codimensional orthogonal projection, \PC\>\C\{tt~l)/n. It is also shown that there is a pathological body, K , all of whose orthogonal projections have volume about \fh~ times as large as |Ä"| . 0. Introduction The problems discussed in this paper concern the areas of shadows (or...
متن کاملReductibility of Convex Bodies
Introduction Let C be a convex bounded body in the n-dimensional Euclidean space E, and let B be its boundary. We use Busemann's notations: E(a,b) for the segment {(1 — p)a + pb: 0 < p ^ 1} from a to b, and S(z, p) for the sphere {x: \\z — x\\ = p). We now recall the definition of Hammer's associated bodies C(r). Let Cb(r) be the set obtained from C by a similitude transformation with a ratio r...
متن کاملOn the volume product of planar polar convex bodies — lower estimates with stability
∗Research (partially) supported by Hungarian National Foundation for Scientific Research, grant nos. K68398, K75016, FP7 IEF grant GEOSUMSETS. ∗∗Research (par-
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 1991
ISSN: 0002-9939
DOI: 10.2307/2048618