arXiv math/98040231998-04-06math.MG

Another low-technology estimate in convex geometry

Greg Kuperberg

备注：: 11 pages
发表信息：: Math. Sci. Res. Inst. Publ. 34 (1999), 117-121

英文摘要：

We give a short argument that for some C > 0, every n-dimensional Banach ball K admits a 256-round subquotient of dimension at least C n/(log n). This is a weak version of Milman's quotient of subspace theorem, which lacks the logarithmic factor.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

A low-technology estimate in convex geometry

Let $K$ be an $n$-dimensional symmetric convex body with $n \ge 4$ and let $K\dual$ be its polar body. We present an elementary proof of the fact that $$(\Vol K)(\Vol K\dual)\ge \frac{b_n^2}{(\log_2 n)^n},$$ where $b_n$ is the volume of the Euclidean ball of radius 1. The inequality is asymptotically weaker than the estimate of Bourgain and Milman, which replaces the $\log_2 n$ by a constant. However, there is no known elementary proof of the Bourgain-Milman theorem.

1992-11-01math.FAmath.MG

Some new positions of maximal volume of convex bodies

In this paper, we extend and generalize several previous works on maximal-volume positions of convex bodies. First, we analyze the maximal positive-definite image of one convex body inside another, and the resulting decomposition of the identity. We discuss continuity and differentiability of the mapping associating a body with its positive John position. We then introduce the saddle-John position of one body inside another, proving that it shares some of the properties possessed by the position of maximal volume, and explain how this can be used to improve volume ratio estimates. We investiga…

2022-07-25math.MG

Yet another short proof of Bourgain's distorsion estimate

We use a self-improvement argument to give a very short and elementary proof of the result of Bourgain saying that regular trees do not admit bi-Lipschitz embeddings into uniformly convex Banach spaces.

2013-02-07math.FAmath.MG

About the continuity of one operation with convex compacts in finite-dimensional normed spaces

In this paper, we study the deformation of the intersection of one compact set with a closed neighborhood of another compact set by changing the radius of this neighborhood. It is shown that in finite-dimensional normed spaces, in the case when both compact sets are non-empty convex subsets, such an operation is continuous in the topology generated by the Hausdorff metric. The question of the continuous dependence of the described intersection on the radius of the neighborhood arose as a by-product of the development of the theory of extremal networks. However, it turned out to be interesting …

2022-11-07math.MG

Pairwise intersecting homothets of a convex body

We show that the maximum number of pairwise intersecting positive homothets of a $d$-dimensional centrally symmetric convex body, none of which contains the center of another in its interior, is at most $3^{d+1}$. Also, we improve upper bounds for cardinalities of $k$-distance sets in Minkowski spaces.

2017-04-01math.COmath.MG

On the shape of a convex body with respect to its second projection body

We prove results relative to the problem of finding sharp bounds for the affine invariant $P(K)=V(ΠK)/V^{d-1}(K)$. Namely, we prove that if $K$ is a 3-dimensional zonoid of volume 1, then its second projection body $Π^2K$ is contained in 8K, while if $K$ is any symmetric 3-dimensional convex body of volume 1, then $Π^2K$ contains 6K. Both inclusions are sharp. Consequences of these results include a stronger version of a reverse isoperimetric inequality for 3-dimensional zonoids-established by the author in a previous work, a reduction for the 3-dimensional Petty conjecture to another isoperim…

2014-09-16math.MG

更多相关论文收起更多论文

The mean width of random polytopes circumscribed around a convex body

Let K be a d-dimensional convex body, and let $K^{(n)}$ be the intersection of n halfspaces containing $K$ whose bounding hyperplanes are independent and identically distributed. Under suitable distributional assumptions, we prove an asymptotic formula for the expectation of the difference of the mean widths of $K^{(n)}$ and K, and another asymptotic formula for the expectation of the number of facets of $K^{(n)}$. These results are achieved by establishing an asymptotic result on weighted volume approximation of $K$ and by "dualizing" it using polarity.

2009-01-22math.MGmath.PR

Complex Geodesics on Convex Domains

Existence and uniqueness of complex geodesics joining two points of a convex bounded domain in a Banach space $X$ are considered. Existence is proved for the unit ball of $X$ under the assumption that $X$ is 1-complemented in its double dual. Another existence result for taut domains is also proved. Uniqueness is proved for strictly convex bounded domains in spaces with the analytic Radon-Nikodym property. If the unit ball of $X$ has a modulus of complex uniform convexity with power type decay at 0, then all complex geodesics in the unit ball satisfy a Lipschitz condition. The results are appl…

2009-07-07math.CVmath.FAmath.MG

A solution to some problems of Conway and Guy on monostable polyhedra

A convex polyhedron is called monostable if it can rest in stable position only on one of its faces. The aim of this paper is to investigate three questions of Conway, regarding monostable polyhedra, which first appeared in a 1969 paper of Goldberg and Guy (M. Goldberg and R.K. Guy, Stability of polyhedra (J.H. Conway and R.K. Guy), SIAM Rev. 11 (1969), 78-82). In this note we answer two of these problems and make a conjecture about the third one. The main tool of our proof is a general theorem describing approximations of smooth convex bodies by convex polyhedra in terms of their static equil…

2021-03-18math.DGmath.DSmath.MG