Related papers: The extremals of Stanley's inequalities for partia…
The Alexandrov-Fenchel inequality, a far-reaching generalization of the classical isoperimetric inequality to arbitrary mixed volumes, lies at the heart of convex geometry. The characterization of its extremal bodies is a long-standing open…
In this paper we introduce an enhanced notion of extremal systems for sets in locally convex topological vector spaces and obtain efficient conditions for set extremality in the convex case. Then we apply this machinery to deriving new…
A classical result of Kahn and Saks states that given any partially ordered set with two distinguished elements, the number of linear extensions in which the ranks of the distinguished elements differ by $k$ is log-concave as a function of…
We prove a Ramsey theorem for finite sets equipped with a partial order and a fixed number of linear orders extending the partial order. This is a common generalization of two recent Ramsey theorems due to Soki\'c. As a bonus, our proof…
The note is devoted to estimates for convolutions appearing in some class of stochastic Volterra equations. Two maximal inequalities and exponential tail estimate are proved by the fractional method of infinite dimensional stochastic…
A necessary and sufficient condition on the precompactness of extremal sequences for one dimensional $\alpha$-Strichartz inequalities, equivalently $\alpha$-Fourier extension estimates, is established based on the profile decomposition…
The paper explores a new extremality model involving collections of arbitrary families of sets. We demonstrate its applicability to set-valued optimization problems with general preferences, weakening the assumptions of the known results…
Stanley introduces polynomials which help evaluate symmetric group characters and conjectures that the coefficients of the polynomials are positive. Stanley later gives a conjectured combinatorial interpretation for the coefficients of the…
We give a broad survey of inequalities for the number of linear extensions of finite posets. We review many examples, discuss open problems, and present recent results on the subject. We emphasize the bounds, the equality conditions of the…
In this article, after recalling and discussing the conventional extremality, local extremality, stationarity and approximate stationarity properties of collections of sets and the corresponding (extended) extremal principle, we focus on…
We survey some of the mechanisms used to prove that naturally defined sequences in combinatorics are log-concave. Among these mechanisms are Alexandrov's inequality for mixed discriminants, the Alexandrov Fenchel inequality for mixed…
In this note we establish some appropriate conditions for stochastic equality of two random variables/vectors which are ordered with respect to convex ordering or with respect to supermodular ordering. Multivariate extensions of this result…
In this paper we develop new extremal principles in variational analysis that deal with finite and infinite systems of convex and nonconvex sets. The results obtained, unified under the name of tangential extremal principles, combine primal…
We study the number of linear extensions of a partial order with a given proportion of comparable pairs of elements, and estimate the maximum and minimum possible numbers. We also consider a random interval partial order on $n$ elements,…
The conventional definition of extremality of a finite collection of sets is extended by replacing a fixed point (extremal point) in the intersection of the sets by a collection of sequences of points in the individual sets with the…
The core arguments used in various proofs of the extremal principle and its extensions as well as in primal and dual characterizations of approximate stationarity and transversality of collections of sets are exposed, analyzed and refined,…
For any lattice polytope $P$, we consider an associated polynomial $\bar{\delta}_{P}(t)$ and describe its decomposition into a sum of two polynomials satisfying certain symmetry conditions. As a consequence, we improve upon known…
We establish several independent results concerning extremal, left modular, congruence uniform, and semidistributive lattices. An equivalent characterization of left modular lattices is obtained in terms of edge-labellings, together with…
The paper proposes another extension of the extremal principle. A new extremality model involving collections of arbitrary families of sets is studied. It generalizes the conventional model based on linear translations of given sets as well…
We investigate the existence of extremals for Hardy-Sobolev inequalities involving the Dirichlet fractional Laplacian of order s, 0<s<1, on half-spaces.