We exhibit efficient algorithms to perform the following task: Given a function f defined on a finite subset E ⊂ R n , compute a C m function F on R n , with a controlled C m norm, that approximates f on the subset E.
We show that there exists a sequence ε n ց 0 for which the following holds: Let K ⊂ R n be a compact, convex set with a non-empty interior. Let X be a random vector that is distributed uniformly in K. Then there exist a unit vector θ in R n , t 0 ∈ R and σ > 0 such thatwhere the supremum runs over all measurable sets A ⊂ R, and where ·, · denotes the usual scalar product in R n . Furthermore, under the additional assumptions that the expectation of X is zero and that the covariance matrix of X is the identity matrix, we may assert that most unit vectors θ satisfy ( * ), with t 0 = 0 and σ = 1. Corresponding principles also hold for multidimensional marginal distributions of convex sets.
Let L(f) denote the Legendre transform of a function f: ℝn → ℝ. A theorem of K. Ball about even functions is generalized, and it is proved that, for any measurable function f ≥ 0, there exists a translation f(x) = f(x−a) such that
∫ℝne−truef˜∫ℝne−scriptL(truef˜)⩽(2π)n.
Let K ⊂ R n be a convex body and ε > 0. We prove the existence of another convex body K ⊂ R n , whose Banach-Mazur distance from K is bounded by 1 + ε, such that the isotropic constant of K is smaller than c/ √ ε, where c > 0 is a universal constant. As an application of our result, we present a slight improvement on the best general upper bound for the isotropic constant, due to Bourgain.
The localization technique from convex geometry is generalized to the setting of Riemannian manifolds whose Ricci curvature is bounded from below. In a nutshell, our method is based on the following observation: When the Ricci curvature is non-negative, log-concave measures are obtained when conditioning the Riemannian volume measure with respect to an integrable geodesic foliation. The Monge mass transfer problem plays an important role in our analysis.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.