We discuss some key results from convex analysis in the setting of topological groups and monoids. These include separation theorems, Krein-Milman type theorems, and minimax theorems.

Abstract John’s fundamental theorem characterizing the largest volume ellipsoid contained in a convex body $K$ $\mathbb{R}^{d}$ has seen several generalizations and extensions. One direction, initiated by V. Milman is to replace ellipsoids positions (affine images) of another $L$. Another, more recent direction consider logarithmically concave functions on instead bodies: we designate some spec...

A consequence of the celebrated Dvoretzky–Milman theorem is a strong law large numbers for d-dimensional random projections n-dimensional cube. It shows that, with respect to Hausdorff distance, uniform projection cube [−1/n,+1/n]n onto Rd converges almost surely centered Euclidean ball radius 2/π, as n→∞. We start by providing an alternative proof this via Artstein–Vitale compact sets. Then, e...

0. Introduction We work over an algebraically closed eld k of characteristic 0. 0.1. Statement. In this paper, we use techniques of toric geometry to reprove the following theorem: Theorem 0.1. Let X be a projective variety of nite type over k, and let Z X be a proper closed subset. Let G Aut k (Z X) be a nite group. Then there is a G-equivariant modii-cation r : X 1 ! X such that X 1 is nonsin...

We extend recent work on nonlinear optimal control problems with integer restrictions on some of the control functions (mixed-integer optimal control problems, MIOCP). We improve a theorem [25] that states that the solution of a relaxed and convexified problem can be approximated with arbitrary precision by a solution fulfilling the integer requirements. Unlike in previous publications the new ...

This note is devoted to the study of the dependence on p of the constant in the reverse Brunn-Minkowski inequality for p-convex balls (i.e. p-convex symmetric bodies). We will show that this constant is estimated as c ≤ C(p) ≤ C , for absolute constants c > 1 and C > 1. Let K ⊂ IR n and 0 < p ≤ 1. K is called a p-convex set if for any λ, μ ∈ (0, 1) such that λ + μ = 1 and for any points x, y ∈ ...

We present the current state of the art concerning the global and local Markov inequalities in the complex plane. This paper is based on a talk given during the Workshop on Multivariate Approximation in honor of Prof. Len Bos 60th birthday, and rests on articles [4], [5] and [6]. Our research is inspired by papers by Bos and Milman where global and local Markov inequalities are compared in the ...

— We give a proof, based on the Poincaré inequality, of the symmetric property (τ) for the Gaussian measure. If f :Rd → R is continuous, bounded from below and even, we define Hf(x) = infy f(x+y)+ 1 2 |y|2 and we have ∫ e−f dγd ∫ e dγd 1. This property is equivalent to a certain functional form of the BlaschkeSantaló inequality, as explained in a paper by Artstein, Klartag and Milman. RÉSUMÉ. —...

Abstract We establish the local-to-global property of synthetic curvature-dimension condition for essentially non-branching locally finite metric-measure spaces, extending work [Cavalletti and Milman in Invent Math 226(1):1–137, 2021].

Banach spaces have been given by Edelstein but under hypotheses on the existence of strong limit points for the sequence I Unx I which are difficult to verify. Cf. Edelstein, M., "On fixed and periodic points under contractive mappings," J. London Math. Soc., 37, 74-79 (1962), "On nonexpansive mappings of Banach spaces," Proc. Cambridge Philos. Soc., 60, 439-447 (1964). 9 Added in proof: After ...