In this paper we will survey the the various forms of convexity in symplectic geometry, paying particular attention to applications of convexity in low dimensional topology.

This paper provides some results on different types of convexity and concavity in the class of multivariate copulas. We also study their properties and provide several examples to illustrate our results.

Often objects which are not convex in the mathematical sense are treated as`perceptually convex'. We present an algorithm for recognition of the perceptual convexity of a 2-d contour. We start by reducing the notion ofà contour is perceptually convex' to the notion ofà contour is Y-convex'. The latter reeects an absence of large concavities in the OY direction of an XOY frame. Then we represent...

The main purpose of this paper is to study normed hypervector spaces. We generalize some definitions such as basis, convexity, operator norm, closed set, Cauchy sequences, and continuity in such spaces and prove some theorems about them.

For x = (x1, x2, · · · , xn) ∈ R+, the symmetric function φn(x, r) is defined by φn(x, r) = φn(x1, x2, · · · , xn; r) = ∏ 1≤i1<i2···<ir≤n  r ∑ j=1 1 + xij xij  , where r = 1, 2, · · · , n, and i1, i2, · · · , in are positive integers. In this article, the Schur convexity, Schur multiplicative convexity and Schur harmonic convexity of φn(x, r) are discussed. As applications, some inequalitie...

We introduce an extension of the standard Local-to-Global Principle used in the proof of the convexity theorems for the momentum map to handle closed maps that take values in a length metric space. This extension is used to study the convexity properties of the cylinder valued momentum map introduced by Condevaux, Dazord, and Molino in [8] and allows us to obtain the most general convexity stat...

We present a proof of strict g-convexity in 2D for solutions generated Jacobian equations with g-Monge–Ampère measure bounded away from 0. Subsequently this implies $$C^1$$ differentiability the case above. Our follows one given by Trudinger and Wang Monge–Ampère case. Thus, like theirs, our argument is local yields quantitative estimate on g-convexity. As result new even optimal transport case...

We study convexity properties of the zeros of some special functions that follow from the convexity theorem of Sturm. We prove results on the intervals of convexity for the zeros of Laguerre, Jacobi and ultraspherical polynomials, as well as functions related to them, using transformations under which the zeros remain unchanged. We give upper as well as lower bounds for the distance between con...

We prove that the formulae of the conjugates of the precomposition with a linear operator, of the sum of finitely many functions and of the sum between a function and the precomposition of another one with a linear operator hold even when the convexity assumptions are replaced by almost convexity or nearly convexity. We also show that the duality statements due to Fenchel hold when the function...