Correction to: Intersection bodies of polytopes
نویسندگان
چکیده
منابع مشابه
COUNTEREXAMPLES TO CONVEXITY OF k-INTERSECTION BODIES
It is a well-known result due to Busemann that the intersection body of an origin-symmetric convex body is also convex. Koldobsky introduced the notion of k-intersection bodies. We show that the k-intersection body of an origin-symmetric convex body is not necessarily convex if k > 1.
متن کاملIntersection Bodies and Valuations
All GL (n) covariant star-body-valued valuations on convex polytopes are completely classified. It is shown that there is a unique nontrivial such valuation. This valuation turns out to be the so-called “intersection operator”—an operator that played a critical role in the solution of the Busemann-Petty problem. Introduction. A function Z defined on the set K of convex bodies (that is, of conve...
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We introduce complex intersection bodies and show that their properties and applications are similar to those of their real counterparts. In particular, we generalize Busemann’s theorem to the complex case by proving that complex intersection bodies of symmetric complex convex bodies are also convex. Other results include stability in the complex Busemann-Petty problem for arbitrary measures an...
متن کاملIsomorphic Properties of Intersection Bodies
We study isomorphic properties of two generalizations of intersection bodies the class I k of k-intersection bodies in R and the class BPk of generalized k-intersection bodies in R. In particular, we show that all convex bodies can be in a certain sense approximated by intersection bodies, namely, if K is any symmetric convex body in R and 1 6 k 6 n− 1 then the outer volume ratio distance from ...
متن کاملThrifty approximations of convex bodies by polytopes
Given a convex body C ⊂ Rd containing the origin in its interior and a real number τ > 1 we seek to construct a polytope P ⊂ C with as few vertices as possible such that C ⊂ τP . Our construction is nearly optimal for a wide range of d and τ . In particular, we prove that if C = −C then for any 1 > > 0 and τ = 1 + one can choose P having roughly −d/2 vertices and for τ = √ d one can choose P ha...
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ژورنال
عنوان ژورنال: Beiträge Zur Algebra Und Geometrie / Contributions To Algebra And Geometry
سال: 2022
ISSN: ['2191-0383', '0138-4821']
DOI: https://doi.org/10.1007/s13366-022-00638-y