Uneven Splitting of Ham Sandwiches

Uneven Splitting of Ham Sandwiches

Let μ1, . . . , μn be continuous probability measures on Rn and α1, . . . , αn ∈ [0, 1]. When does there exist an oriented hyperplane H such that the positive half-space H+ has μi(H) = αi for all i ∈ [n]? It is well known that such a hyperplane does not exist in general. The famous ham sandwich theorem states that if αi = 2 for all i, then such a hyperplane always exists. In this paper we give ...

Ham Sandwiches, Staircases and Counting Polynomials

Buckyball sandwiches

Two-dimensional (2D) materials have considerably expanded the field of materials science in the past decade. Even more recently, various 2D materials have been assembled into vertical van der Waals heterostacks, and it has been proposed to combine them with other low-dimensional structures to create new materials with hybridized properties. We demonstrate the first direct images of a suspended ...

Bipartite Sandwiches

A semideenite relaxation (?) for the problem of nding the maximum number (?) of edges in a complete bipartite subgraph of a bipartite graph ? = (V1 V2; E) is considered. For a large class of graphs, the relaxation is better than the LP-relaxation described in 8]. It is shown that (?) is bounded from above by the Lovv asz theta function (LQ(?)) of the graph LQ(?) related to the line graph of ?. ...

Fault Tolerant Firewall Sandwiches