arXiv subjects
Publications and source records attributed to Srinivas Arun.
Given a set $S \subseteq \mathbb{R}^d$, a hollow polytope has vertices in $S$ but contains no other point of $S$ in its interior. We prove upper and lower bounds on the maximum number of vertices of hollow polytopes whose facets are simplices or whose vertices are in general position. We also obtain relatively tight asymptotic bounds for polytopes which do not contain lattice segments of large length.
A finite family $\mathcal F$ of convex sets is $k$-intersecting in $S \subseteq \mathbb{R}^d$ if the intersection of every subset of $k$ convex sets in $\mathcal F$ contains a point in $S$. The Helly number of $S$ is the minimum $k$, if it exists, such that every $k$-intersecting family contains a point of $S$ in its intersection. In this paper, we improve bounds on the Helly number of product sets of the form $A^d$ for various sets $A \subseteq \mathbb{R}$, including the ``exponential grid'' $A = \{\alpha^n : n \in \mathbb{N}\}$ and sets $A\subseteq \mathbb{Z}$ defined by congruence relations.