A probabilistic approach to the geometry of the ℓᵨⁿ-ball

By Franck Barthe, Olivier Guédon, Shahar Mendelson, Assaf Naor. 2005

245 citations recorded by the source corpus.

This article investigates, by probabilistic methods, various geometric questions on B n p , the unit ball of ln p . We propose realizations in terms of independent random variables of several distributions on B n p , including the normalized volume measure. These representations allow us to unify and extend the known results of the sub-independence of coordinate slabs in B n p . As another application, we compute moments of linear functionals on B n p , which gives sharp constants in Khinchine's inequalities on B n p and determines the 2-constant of all directions on B n p . We also study the extremal values of several Gaussian averages on sections of B n p (including mean width and l-norm), and derive several monotonicity results as p varies. Applications to balancing vectors in l 2 and to covering numbers of polyhedra complete the exposition.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7de0e5acab2361cad5f5c709854e97e9f407b6e9

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Empirical Processes with a Bounded Ψ1 Diameter (2010, background): “There are isotropic bodies with a subgaussian volume measure – for example, isotropic positions of Bn p for p ≥ 2 [5].”
The power of localization for efficiently learning linear separators with noise (2013, background): “, xd) is sampled uniformly from Sd−1, then the distribution of (x1, x2) is log-concave (see Corollary 4 of [BGMN05]).”
On the infimum convolution inequality (2008, background): “Lemma 6 from [3] gives (after rescaling by rp,n),”
Existence of solutions to the even dual Minkowski problem (2018, background): “It is still being intensively studied, see, for example, Barthe, Guédon, Mendelson & Naor [4], Böröczky & Henk [6], Böröczky-LYZ [8, 9, 10], Henk & Linke [31], Ludwig [38], Ludwig & Reitzner [40],”
Restricted Isometry Property of Matrices with Independent Columns and Neighborly Polytopes by Random Sampling (2009, methodology): “ytopes K+(A) and K(A) are m-neighborly and m-centrally-neighborly, respectively, whenever m≤ cn log2(CN/n), where C,c>0 are universal constants and c1 is given in Lemma 2.6. Remark: It is known ([2]) that there is a universal constant ψsuch that the uniform probability measure on the ball {x∈ Rn”
On the L dual Minkowski problem (2018, background): “See, for example, Barthe, Guédon, Mendelson & Naor [4], Böröczky & Henk [7], Böröczky, Lutwak, Yang & Zhang [9–11], Henk & Linke [29], Ludwig [44], Ludwig & Reitzner [47], Naor [54], Naor & Romik [55], Paouris & Werner [56], Stancu [62,63], Xiong [65], Zhu [70], and Zou & Xiong [73].”
The logarithmic Minkowski problem for polytopes (2014, methodology): “[57] Naor, A.: The surface measure and cone measure on the sphere of lnp .”