A theorem of Tietze and Nakamija [Ti, N], from 1928, asserts that if a subset X of R is closed, connected, and locally convex, then it is convex. There are many generalizations of this “local to global convexity” phenomenon in the literature. See, e.g., [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 is repla...

Poisson Lie groups appeared in the work of Drinfel'd (see, e.g., [Drl, Dr2]) as classical objects corresponding to quantum groups. Going in the other direction, we may say that a Poisson Lie group is a group of symmetries of a phase space that are allowed to "twist," in a certain sense, the symplectic or Poisson structure. The Poisson structure on the group controls this twisting in a precise w...

One of the fundamental results regarding intersection bodies is Busemann’s theorem, which states that the intersection body of a symmetric convex body is convex. Thus, it is natural to ask how much of convexity is guaranteed under the intersection body operation. In this paper we provide several results about the (strict) improvement of convexity for intersection bodies of symmetric bodies of r...

We show that a piecewise-linear (PL) complete immersion of a connected manifold of dimension n−1 into n-dimensional Euclidean space (n > 2) is the boundary of a convex polyhedron, bounded or unbounded, if and only if the interior of each (n− 3)-face has a point with a neighborhood (on the surface) that lies on the boundary of a convex body. No initial assumptions about the topology or orientabi...

For a function defined on a convex set in a Euclidean space, midpoint convexity is the property requiring that the value of the function at the midpoint of any line segment is not greater than the average of its values at the endpoints of the line segment. Midpoint convexity is a well-known characterization of ordinary convexity under very mild assumptions. For a function defined on the integer...

The main results of this paper offer sufficient conditions in order that an approximate lower Hermite–Hadamard type inequality implies an approximate Jensen convexity property. The key for the proof of the main result is a Korovkin type theorem.

“Theorem of the alternative” is the generic name of different results used as an important tool in optimization. Though the early versions have been obtained under strong conditions of convexity, for the further extensions, these conditions have been weakened using generalized convexity beyond vector space structure. This is the case of the result proved by Jeyakumar [4], who used an extension ...

In this paper we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X, d,m) which is stable under measured Gromov-Hausdorff convergence and rules out Finsler geometries. It can be given in terms of an enforcement of the Lott, Sturm and Villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow. Besides stability, i...

Abstract Motivated by the characterization theorem about Jensen convexity of quasiarithmetic means obtained authors in 2021, our main goal is to establish a quasideviation as well Bajraktarević without any additional and unnatural regularity assumptions .