Topological Drawings Meet Classical Theorems from Convex Geometry

Theorems from the interface of convex geometry and analysis

M-FUZZIFYING TOPOLOGICAL CONVEX SPACES

The main purpose of this paper is to introduce the compatibility of $M$-fuzzifying topologies and $M$-fuzzifying convexities, define an $M$-fuzzifying topological convex space, and give a method to generate an $M$-fuzzifying topological convex space. Some characterizations of $M$-fuzzifying topological convex spaces are presented. Finally, the notion of $M$-fuzzifying weak topologies is obtaine...

Convex Solutions of Elliptic Differential Equations in Classical Differential Geometry

The issue of convexity is fundamental in the theory of partial differential equations. We discuss some recent progress of convexity estimates for solutions of nonlinear elliptic equations arising from some classical problems in differential geometry. We first review some works in the literature on the convexity of solutions of quasilinear elliptic equations in Rn. The study of geometric propert...

Optimal Morphs of Convex Drawings

We give an algorithm to compute a morph between any two convex drawings of the same plane graph. The morph preserves the convexity of the drawing at any time instant and moves each vertex along a piecewise linear curve with linear complexity. The linear bound is asymptotically optimal in the worst case. 1998 ACM Subject Classification G.2.2. Graph Theory

Convex-Arc Drawings of Pseudolines

Introduction. A pseudoline is formed from a line by stretching the plane without tearing: it is the image of a line under a homeomorphism of the plane [13]. In arrangements of pseudolines, pairs of pseudolines intersect at most once and cross at their intersections. Pseudoline arrangements can be used to model sorting networks [1], tilings of convex polygons by rhombi [4], and graphs that have ...