On (α,p)-convex contraction and asymptotic regularity

Regularity estimates for convex multifunctions

The main result of the paper contains an exact formula for the rate of regularity of a set-valued mapping with a convex graph. As a consequence we find an exact expression for the rate of regularity of a set-valued mapping associated with so called constraint system. It turns out that the rate is equal to the upper bound of Robinson-type estimates over all norms in the graph space of the homoge...

Counterexamples to Elliptic Regularity and Convex Integration

The purpose of this article is to give a survey of recent results on the construction of elliptic equations and systems with critical regularity properties. The constructions are based on the method of convex integration, combined with a careful analysis of oscillations in the spirit of compensated compactness. Our aim is to emphasize the approach which separates the analysis of oscillations fr...

On the Asymptotic Linearity of Castelnuovo-mumford Regularity

Contraction and Expansion of Convex Sets

Helly’s theorem is one of the fundamental results in discrete geometry [9]. It states that if every 6 d+1 sets in a set system S of convex sets in R have non-empty intersection then all of the sets in S have non-empty intersection. Equivalently, if the entire family S has empty intersection, then there is a subset S ′ ⊂ S (a witness) of size 6 d+1 which also has empty intersection. Over the yea...

Convex Hull Asymptotic Shape Evolution

The asymptotic properties of Rapidly exploring Random Tree (RRT) growth in large spaces is studied both in simulation and analysis. The main phenomenon is that the convex hull of the RRT reliably evolves into an equilateral triangle when grown in a symmetric planar region (a disk). To characterize this and related phenomena from flocking and swarming, a family of dynamical systems based on incr...