After observing that the well-known convexity theorems of symplectic geometry also hold for compact contact manifolds we use this fact together with a recent symplectic orbifold version of Delzant’s theorem due to Lerman and Tolman [LT] to show that every compact toric contact manifold can be obtained by a contact reduction from an odd dimensional sphere.

This is an expository paper based on Border [1]. Starting from basic convexity results, I present a proof of the so-called Equilibrium Theorem, which states the existence of a free disposal equilibrium price vector in an Arrow-Debreu economy with a continuous excess demand function.

A theorem of Tietze and Nakajima, from 1928, asserts that if a subset X of R is closed, connected, and locally convex, then it is convex [Ti, N]. There are many generalizations of this “local to global convexity” phenomenon in the literature; a partial list is [BF, C, Ka, KW, Kl, SSV, S, Ta]. This paper contains an analogous “local to global convexity” theorem when the inclusion map of X to R i...

We apply an existence theorem of variational inclusion problem on metric spaces to study optimization problems, set-valued vector saddle point problems, bilevel problems, and mathematical programs with equilibrium constraint on metric spaces. We study these problems without any convexity and compactness assumptions. Our results are different from any existence results of these types of problems...

This paper provides a succinct proof of a 1973 theorem of Lieb that establishes the concavity of a certain trace function. The development relies on a deep result from quantum information theory, the joint convexity of quantum relative entropy, as well as a recent argument due to Carlen and Lieb.

The purpose of this paper is to establish an existence result for nonconvex variational problems with Bochner integral constraints in separable Asplund spaces via the Euler–Lagrange inclusion, under the saturation hypothesis on measure spaces, which makes the Lyapunov convexity theorem valid in Banach spaces. The approach is based on the indirect method of the calculus of variations.

The object of the present paper is to derive a property of certain meromorphic functions in the punctured unit disk. Our main theorem contains certain sufficient conditions for starlikeness and close-to-convexity of order α of meromorphic functions. 2000 Mathematics Subject Classification. 30C45.

Abstract The main result is an equality type mean value theorem for tangentially convex functions in terms of tangential subdifferentials, which generalizes the classical one differentiable functions, as well Wegge functions. new then applied, analogously to what done case, characterize, context, Lipschitz increasingness with respect ordering induced by a closed cone, convexity, and quasiconvex...

The subject of these lecture notes is comparison theory in Riemannian geometry: What can be said about a complete Riemannian manifold when (mainly lower) bounds for the sectional or Ricci curvature are given? Starting from the comparison theory for the Riccati ODE which describes the evolution of the principal curvatures of equidistant hypersurfaces, we discuss the global estimates for volume a...