We generalize a well-known result of L. Caffarelli on Lipschitz estimates for optimal transportation T between uniformly log-concave probability measures. Let T : R → R be an optimal transportation pushing forward μ = e dx to ν = e dx. Assume that 1) the second differential quotient of V can be estimated from above by a power function, 2) modulus of convexity of W can be estimated from below by...

the notion of a bead metric space is defined as a nice generalization of the uniformly convex normed space such as $cat(0)$ space, where the curvature is bounded from above by zero. in fact, the bead spaces themselves can be considered in particular as natural extensions of convex sets in uniformly convex spaces and normed bead spaces are identical with uniformly convex spaces. in this paper, w...

In the present paper, we first prove the logarithmic convexity of the elementary function b x −a x x , where x 6= 0 and b > a > 0. Basing on this, we then provide a simple proof for Schur-convex properties of the extended mean values, and, finally, discover some convexity related to the extended mean values.

Khrapchenko’s classical lower bound n2 on the formula size of the parity function f can be interpreted as designing a suitable measure of sub-rectangles of the combinatorial rectangle f−1(0) × f−1(1). Trying to generalize this approach we arrived at the concept of convex measures. We prove the negative result that convex measures are bounded by O(n2) and show that several measures considered fo...

In this paper, we define the notions of fuzzy congruence relations and fuzzy convex subalgebras on a commutative residuated lattice and we obtain some related results. In particular, we will show that there exists a one to one correspondence between the set of all fuzzy congruence relations and the set of all fuzzy convex subalgebras on a commutative residuated lattice. Then we study fuzzy...

knowing the fact that the main weakness of the most standard methods including k-means and hierarchical data clustering is their sensitivity to initialization and trapping to local minima, this paper proposes a modification of convex data clustering  in which there is no need to  be peculiar about how to select initial values. due to properly converting the task of optimization to an equivalent...

The main purpose of this paper is to introduce the compatibility of $M$-fuzzifying topologies and $M$-fuzzifying convexities, define an $M$-fuzzifying topological convex space, and give a method to generate an $M$-fuzzifying topological convex space. Some characterizations of $M$-fuzzifying topological convex spaces are presented. Finally, the notion of $M$-fuzzifying weak topologies is obtaine...

In this note, we prove the following result. Let K ⊂ Rd be a convex body with the origin O in its interior. If there is a number λ ∈ (0, 1) such that the n-dimensional volume of the convex hull of the union of K with the translates of λK , by a vector x , depends only on the Euclidean norm of x , then K is a Euclidean ball.