11 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
Difference sets in
Let be a natural number. We show that for any sufficiently large finite subset
of that is not contained in a translate of a hyperplane. By
a construction of Stanchescu, this is best possible and thus resolves an old
question first raised by Uhrin.Comment: 15 page
Inverse Additive Problems for Minkowski Sumsets II
The Brunn-Minkowski Theorem asserts that for convex bodies , where
denotes the -dimensional Lebesgue measure. It is well-known that
equality holds if and only if and are homothetic, but few
characterizations of equality in other related bounds are known. Let be a
hyperplane. Bonnesen later strengthened this bound by showing where
and
. Standard
compression arguments show that the above bound also holds when
and , where denotes a
projection of onto , which gives an alternative generalization
of the Brunn-Minkowski bound. In this paper, we characterize the cases of
equality in this later bound, showing that equality holds if and only if
and are obtained from a pair of homothetic convex bodies by `stretching'
along the direction of the projection, which is made formal in the paper. When
, we characterize the case of equality in the former bound as well
New lower bounds for cardinalities of higher dimensional difference sets and sumsets
Let d ≥ 4 be a natural number and let A be a finite, non-empty subset of R
d such that A is not contained in a translate of a hyperplane. In this setting, we show that |A−A| ≥ (2d − 2 + 1/d −1) |A| −Od(|A|1−δ), for some absolute constant δ > 0 that only depends on d. This provides a sharp main term, consequently answering questions of Ruzsa and Stanchescu up to an Od(|A|
1−δ) error term. We also prove new lower bounds for restricted type difference sets and asymmetric sumsets in Rd
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