Reuben Neamiah La Haye

• University of California, Davis

This paper makes two contributions to optimization theory derived from new methods of discrete convex analysis. Our first contribution is to stochastic optimization. The scenario approach developed by Calafiore and Campi to attack chance-constrained convex programs (i.e., optimization problems with convex constraints that are parametrized by an unc...

This paper presents a new variation of Tverberg’s theorem. Given a discrete set S of Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{d}$$\end{document},...

We prove variations of Carath\'eodory's, Helly's and Tverberg's theorems where the sets involved are measured according to continuous functions such as the volume or diameter. Among our results, we present continuous quantitative versions of Lov\'asz's colorful Helly theorem, B\'ar\'any's colorful Carath\'eodory's theorem, and the colorful Tverberg...

We study S-convex sets, which are the geometric objects obtained as the intersection of the usual convex sets in ℝd with a proper subset S ⊂ ℝd, and contribute new results about their S-Helly numbers. We extend prior work for S = ℝd, ℤd, and ℤd−k × ℝk, and give some sharp bounds for several new cases: low-dimensional situations, sets that have some...

This paper presents a new variation of Tverberg's theorem. Given a discrete set $S$ of $R^d$, we study the number of points of $S$ needed to guarantee the existence of an $m$-partition of the points such that the intersection of the $m$ convex hulls of the parts contains at least $k$ points of $S$. The proofs of the main results require new quantit...

We prove variations of Carath\'eodory's, Helly's and Tverberg's theorems where the sets involved are measured according to continuous functions such as the volume or diameter. Among our results, we present continuous quantitative versions of Lov\'asz's colorful Helly theorem, B\'ar\'any's colorful Carath\'eodory's theorem, and the colorful Tverberg...

We study $S$-convex sets, which are the geometric objects obtained as the intersection of the usual convex sets in $\mathbb R^d$ with a proper subset $S\subset \mathbb R^d$. We contribute new results about their $S$-Helly numbers. We extend prior work for $S=\mathbb R^d$, $\mathbb Z^d$, and $\mathbb Z^{d-k}\times\mathbb R^k$; we give sharp bounds o...

We present Helly-type theorems where the convex sets are required to intersect a subset $S$ of $\mathbb R^d$. This is a continuation of prior work for $S=\mathbb R^d$, $\mathbb Z^d$, and $\mathbb Z^{d-k}\times\mathbb R^k$ (motivated by mixed-integer optimization). We are particularly interested in the case when $S$ has some algebraic structure, in...

This paper presents sixteen quantitative versions of the classic Tverberg, Helly, & Caratheodory theorems in combinatorial convexity. Our results include measurable or enumerable information in the hypothesis and the conclusion. Typical measurements include the volume, the diameter, or the number of points in a lattice.

We present a Rainbow Ramsey version of the well-known Ramsey-type theorem of Richard Rado. We use techniques from the Geometry of Numbers. We also disprove two conjectures proposed in the literature.