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.

We present a proof of strict g-convexity in 2D for solutions generated Jacobian equations with g-Monge–Ampère measure bounded away from 0. Subsequently this implies $$C^1$$ differentiability the case above. Our follows one given by Trudinger and Wang Monge–Ampère case. Thus, like theirs, our argument is local yields quantitative estimate on g-convexity. As result new even optimal transport case...

A set K of nodes of a graph G is geodesically convex (respectively, monophonically convex) if K contains every node on every shortest (respectively, chordless) path joining nodes in K. We investigate the classes of graphs which are characterized by certain local convexity conditions with respect to geodesic convexity, in particular, those graphs in which balls around nodes are convex, and those...

Based on a classic paper by Ginibre [Commun. Math. Phys. 8, 26 (1968)] it is shown that, whenever Bogoliubov's approximation, that is, the replacement of a(0) and a*(0) by complex numbers in the Hamiltonian, asymptotically yields the right pressure, it also implies the asymptotic equality of |a(0)|(2)/V and a*(0)a(0)/V in symmetry breaking fields, irrespective of the existence or absence of Bos...

We use bivariate C 1 cubic splines to deal with convexity preserving scattered data interpolation problem. Using a necessary and suucient condition on Bernstein-B ezier polynomials, we set the convexity preserving interpolation problem into a quadratically constraint quadratic programming problem. We show the existence of convexity preserving interpolatory surfaces under certain conditions on t...

In this work we investigate the behavior of various geodesic convexity parameters with respect to the Cartesian product operation for graphs. First, we show that the convex sets arising from geodesic convexity in a Cartesian product of graphs are exactly the same as the convex sets arising from the usual binary operation ⊕ for making a convexity space out of the Cartesian product of any two con...

A set of vertices S in a graph is convex if it contains all vertices which belong to shortest paths between vertices in S. The convexity number c(G) of a graph G is the maximum cardinality of a convex set of vertices which does not contain all vertices of G. We prove NP-completeness of the problem to decide for a given bipartite graph G and an integer k whether c(G) ≥ k. Furthermore, we identif...