Convexity Conditions for Non - Locally Convex Lattices

Approximate Convexity and Submonotonicity in Locally Convex Spaces

ar X iv : m at h / 06 06 25 6 v 1 [ m at h . G R ] 1 1 Ju n 20 06 SUPERRIGIDITY , GENERALIZED HARMONIC MAPS AND UNIFORMLY CONVEX SPACES

We prove several superrigidity results for isometric actions on metric spaces satisfying some convexity properties. First, we extend some recent theorems of N. Monod on uniform and certain non-uniform irreducible lattices in products of locally compact groups. Second, we include the proof of an unpublished result on commensurability superrigidity due to Margulis. The proofs rely on certain noti...

Lattices of Convex Sets

If F is a vector space over an ordered division ring, C a convex subset of V and L the lattice of convex subsets of C, then we call L a convexity lattice. We give necessary and sufficient conditions for an abstract lattice to be a convexity lattice in the finite dimensional case.

Convexity Conditions for Parameterized Surfaces

Based on a geometrical method, the internal relationships between locally parameterized curves and the local parameterized surfaces are analyzed. A necessary and sufficient condition is derived for the local convexity of parameterized surfaces and functional surfaces. A criterion for local convexity (concavity) of parameterized surfaces is found, also, the criterion condition of binary function...

Fixed Points and Selections of Set-valued Maps on Spaces with Convexity

We provide theorems extending both Kakutani and Browder fixed points theorems for multivalued maps on topological vector spaces, as well as some selection theorems. For this purpose we introduce convex structures more general than those of locally convex and non-locally convex topological vector spaces or generalized convexity structures due to Michael, van de Vel, and Horvath.