Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0612517

On the hyperplane conjecture for random convex sets

Bo'az Klartag·Gady Kozma
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let N > n, and denote by K the convex hull of N independent standard gaussian random vectors in an n-dimensional Euclidean space. We prove that with high probability, the isotropic constant of K is bounded by a universal constant. Thus we verify the hyperplane conjecture for the class of gaussian random polytopes.

Explore related subjects

math.MGmath.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Bo'az Klartag, Gady Kozma. 2006-12-18. On the hyperplane conjecture for random convex sets. https://arxiv.org/abs/math/0612517

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Minimal Covering Bodies and a Minkowski-Type Criterion for Lattice Coverings

We study convex bodies whose translates by a fixed lattice cover space and for which every proper convex subbody loses this property. In three dimensions, the convex hull of independent translates of the six Kuhn tetrahedra always gives a lattice covering. We give a geometric proof using auxiliary tetrahedra and the parity of the covering multiplicity. We also prove polytopality and boundary restrictions for minimal covering bodies, and show that every three-dimensional parallelohedron admits a Kuhn representation with respect to its face-to-face tiling lattice. Constructions from Reeve tetrahedra give non-symmetric minimal covering bodies with eight vertices in dimension three and centrally symmetric ones with sixteen vertices in dimension four, with unbounded volumes for the integer covering lattice. They have pairwise distinct arithmetic contact types, which record lattice contacts and the faces containing them. The covering theorem also gives a finite intersection criterion for a prescribed lattice basis, with three intersection tests in the centrally symmetric case.

math.MG↗
arXiv2026

The Spherical Hadwiger Theorem

We prove the spherical Hadwiger classification in every dimension. For $n\geq1$, every continuous $SO(n+1)$-invariant valuation on the space of all closed spherical convex sets in $\mathbb{S}^n$ can be written uniquely as a linear combination of the spherical intrinsic volumes $V_0,\ldots,V_n$. The same classification holds when the domain is restricted to sets contained in an open hemisphere. The proof is inductive: it reduces the problem to a vanishing statement for simple valuations and uses a continuous alternating cocycle on tuples of spherical points together with a signed coning transform. Through the cone--sphere correspondence, this yields, for $d\geq2$, the corresponding classification of continuous $SO(d)$-invariant conic valuations on all closed convex cones in $\mathbb{R}^d$, without imposing normalization at the zero cone. In particular, $SO$-invariance implies $O$-invariance in both settings.

math.MG↗
arXiv2026

The topology of Gromov--Hausdorff space

We prove that the Gromov--Hausdorff space is homeomorphic to the Hilbert space. This paper is divided into four parts. In Part I, we construct an assignment of a full-support probability measure to every nonempty compact metric space that respects isometries and is continuous for simultaneous Hausdorff convergence of the spaces and weak convergence of the measures. In Part II, we use these measures to construct finite-dimensional local models whose induced pseudometrics approximate the original distances uniformly and whose norms and point maps vary continuously up to orthogonal changes of coordinates. In Part III, we use the local models to prove that the Gromov--Hausdorff space is an absolute retract for all metrizable spaces. In Part IV, we establish a discrete approximation property and conclude that the space of isometry classes of nonempty compact metric spaces is homeomorphic to the real separable infinite-dimensional Hilbert space.

math.MG↗