3 research outputs found
Triangulations and a discrete Brunn-Minkowski inequality in the plane
For a set of points in the plane, not all collinear, we denote by the number of triangles in any triangulation of ; that is, where and are the numbers of points of in the
boundary and the interior of (we use to denote "convex hull of
"). We conjecture the following analogue of the Brunn-Minkowski inequality:
for any two point sets one has
We prove this conjecture in several cases: if , if ,
if , or if none of or has interior points.Comment: 30 page
Sets in with doubling are near convex progressions
For sufficiently small and with , we show either is covered by parallel
hyperplanes, or satisfies , where is the smallest convex
progression (convex set intersected with a sublattice) containing . This
generalizes the Freiman-Bilu theorem, Freiman's theorem, and
recent sharp stability results of the present authors for sumsets in
conjectured by Figalli and Jerison.Comment: 53 pages, comments welcome