Abstract
AbstractWe provide a characterization of those log-concave distributions in $${{\mathbb {R}}}^n$$
R
n
that are contoured distributions, through the $$K_p$$
K
p
-bodies of the distribution, defined by Ball. Our method uses the logarithmic integral for the solution of a Bernstein type approximation problem. In the second part of the paper we state a question for contoured distributions that would provide an alternative approach to the isotropic constant problem.
Publisher
Springer Science and Business Media LLC
Reference18 articles.
1. Artstein-Avidan, S., Giannopoulos, A., Milman, V.D.: Asymptotic Geometric Analysis Mathematical Surveys and Monographs, 202, vol. I. American Mathematical Society, Providence (2015)
2. Ball, K.M.: Logarithmically concave functions and sections of convex sets in $${\mathbb{R} }^n$$. Stud. Math. 8(8), 69–84 (1988)
3. Borell, C.: Convex measures on locally convex spaces. Ark. Mat. 12, 239–252 (1974)
4. Borichev, A.: On the closure of polynomials in weighted spaces of functions on the real line. Indiana Univ. Math. J. 50(2), 829–846 (2001)
5. Bourgain, J.: On the distribution of polynomials on high-dimensional convex sets aspects of functional analysis (Lindenstrauss-Milman eds). Geom. Lecture Notes in Math. 1469, 127–137 (1991)