Related papers: A convex set with a rich difference

We give a construction of a convex set $A \subset \mathbb R$ with cardinality $n$ such that $A-A$ contains a convex subset with cardinality $\Omega (n^2)$. We also consider the following variant of this problem: given a convex set $A$, what…

Suppose that $A$ is a finite, nonempty subset of a cyclic group of either infinite or prime order. We show that if the difference set $A-A$ is ``not too large'', then there is a nonzero group element with at least as many as…

We consider the question of which nonconvex sets can be represented exactly as the feasible sets of mixed-integer convex optimization problems. We state the first complete characterization for the case when the number of possible integer…

A set A of positive integers is called a perfect difference set if every nonzero integer has an unique representation as the difference of two elements of A. We construct dense perfect difference sets from dense Sidon sets. As a consequence…

In this paper we explicitly estimate the number of points in a subset $A \subset \R^{d}$ as a function of the maximum angle $\angle A$ that any three of these points form, provided $\angle A < \theta_d := \arccos(-\frac 1 {d}) \in…

Let K be a simplicial complex with vertex set V = {v_1,..., v_n}. The complex K is d-representable if there is a collection {C_1,...,C_n} of convex sets in R^d such that a subcollection {C_{i_1},...,C_{i_j}} has a nonempty intersection if…

The convexity number of a set $X \subset \mathbb{R}^2$ is the minimum number of convex subsets required to cover it. We study the following question: what is the largest possible convexity number $f(n)$ of $\mathbb{R}^2 \setminus S$, where…

We show that for any $t>1$, the set of unconditional convex bodies in $\mathbb{R}^n$ contains a $t$-separated subset of cardinality at least $\exp \exp (C(t) n)$. This implies that there exists an unconditional convex body in $\mathbb{R}^n$…

Four constructions result from a desire to create enhancements to Cantor's infinite real set cardinality. Each continues to keep Cantor's cardinality formulation in place while providing new comparisons of arbitrary infinite sets. To…

We present a versatile construction allowing one to obtain pairs of integer sets with infinite symmetric difference, infinite intersection, and identical representation functions.

We show that the maximum cardinality of an anti-chain composed of intersections of a given set of n points in the plane with half-planes is close to quadratic in n. We approach this problem by establishing the equivalence with the problem…

Let W be a polyconvex function defined on the 2 x 2 real matrices. The Busemann representative f, say, of W is the largest possible convex representative of W. Writing L for the set of affine functions on R^{5} such that a(A, det A) is less…

A set of reals A={a_1,...,a_2} is called convex if a_{i+1} - a_i > a_i - a_{i-1} for all i. We prove, in particular, that |A-A| \gg |A|^{8/5} \log{-2/5} |A|.

The main objective of this paper is to show that the complement of a rational convex set in $\mathbb{C}^n$ is (n-2)-connected for n>2.

This paper gives two different proofs to a structural theorem of decreasing minimization (lexicographic optimization) on integrally convex sets. The theorem states that the set of decreasingly minimal elements of an integrally convex set…

The aim of this paper is to prove that if a planar set $A$ has a difference set $\Delta(A)$ satisfying $\Delta(A)\subset \Z^++s$ for suitable $s$ than $A$ has at most 3 elements. This result is motivated by the conjecture that the disk has…

We prove that for any collection F of $n \ge 2$ pairwise disjoint compact convex sets in the plane there is a pair of sets A and B in F such that any line that separates A from B separates either A or B from a subcollection of F with at…

We prove a lemma that is useful to get upper bounds for the number of partitions without a given subsum. From this we can deduce an improved upper bound for the number of sets represented by the (unrestricted or into unequal parts)…

It is shown that for a finite set $A$ of four or more complex numbers, the cardinality of the set $C[A]$ of all cross-ratios generated by quadruples of pair-wise distinct elements of $A$ is $|C[A]|\gg…

We construct finite sets of real numbers that have a small difference set and strong local properties. In particular, we construct a set $A$ of $n$ real numbers such that $|A-A|=n^{\log_2 3}$ and that every subset $A'\subseteq A$ of size…