Convexity of Hypersurfaces in Spherical Spaces

Convexity and Geodesic Metric Spaces

In this paper, we first present a preliminary study on metric segments and geodesics in metric spaces. Then we recall the concept of d-convexity of sets and functions in the sense of Menger and study some properties of d-convex sets and d-convex functions as well as extreme points and faces of d-convex sets in normed spaces. Finally we study the continuity of d-convex functions in geodesic metr...

$L_1$-Biharmonic Hypersurfaces in Euclidean Spaces with Three Distinct Principal Curvatures

Chen's biharmonic conjecture is well-known and stays open: The only biharmonic submanifolds of Euclidean spaces are the minimal ones. In this paper, we consider an advanced version of the conjecture, replacing $Delta$ by its extension, $L_1$-operator ($L_1$-conjecture). The $L_1$-conjecture states that any $L_1$-biharmonic Euclidean hypersurface is 1-minimal. We prove that the $L_1$-conje...

Approximate Convexity and Submonotonicity in Locally Convex Spaces

An efficient local approach to convexity testing of piecewise-linear hypersurfaces

We show that a closed piecewise-linear hypersurface immersed in R (n ≥ 3) is the boundary of a convex body if and only if every point in the interior of each (n− 3)-face has a neighborhood that lies on the boundary of some convex body; no assumptions about the hypersurface’s topology are needed. We derive this criterion from our generalization of Van Heijenoort’s (1952) theorem on locally conve...

On difference sequence spaces defined by Orlicz functions without convexity

In this paper, we first define spaces of single difference sequences defined by a sequence of Orlicz functions without convexity and investigate their properties. Then we extend this idea to spaces of double sequences and present a new matrix theoretic approach construction of such double sequence spaces.