A convex polygon as a discrete plane curve.
A survey of Sylvester's problem and its generalizations.
A tight lower bound for convexly independent subsets of the Minkowski sums of planar point sets.
An Upper Bound for the Minimum Diameter of Integral Point Sets.
Another abstraction of the Erdős-Szekeres happy end theorem.
Answer to one of Fishburn's questions: ``Isosceles planar subsets'' [Discrete Comput. Geom. 19 (1998), no. 3, 391--398]
Our short note gives the affirmative answer to one of Fishburn’s questions.
Convexly independent subsets of the Minkowski sum of planar point sets.
Countable Decompositions of R2 and R3 .
Cutting Disjoint Disks by Straight Lines.
Explicit Ramsey graphs and Erdős distance problems over finite Euclidean and non-Euclidean spaces.
Fourier analysis, linear programming, and densities of distance avoiding sets in
We derive new upper bounds for the densities of measurable sets in which avoid a finite set of prescribed distances. The new bounds come from the solution of a linear programming problem. We apply this method to obtain new upper bounds for measurable sets which avoid the unit distance in dimensions . This gives new lower bounds for the measurable chromatic number in dimensions . We apply it to get a short proof of a variant of a recent result of Bukh which in turn generalizes theorems of Furstenberg,...
Halving point sets.
Isosceles sets.
Large convexly independent subsets of Minkowski sums.
Mathematical snapshots from the computational geometry landscape.
On the number of intersections of two polygons
We study the maximum possible number of intersections of the boundaries of a simple -gon with a simple -gon in the plane for . To determine the number is quite easy and known when or is even but still remains open for and both odd. We improve (for ) the easy upper bound to and obtain exact bounds for
On the set of distances between two sets over finite fields.
On the structure of sets with many k-term arithmetic progressions
Optimal bounds for the colored Tverberg problem
We prove a “Tverberg type” multiple intersection theorem. It strengthens the prime case of the original Tverberg theorem from 1966, as well as the topological Tverberg theorem of Bárány et al. (1980), by adding color constraints. It also provides an improved bound for the (topological) colored Tverberg problem of Bárány & Larman (1992) that is tight in the prime case and asymptotically optimal in the general case. The proof is based on relative equivariant obstruction theory.
Sets with large Borsuk number.