We prove several exact quantitative versions of Helly's and Tverberg's theorems, which guarantee that a finite family of convex sets in R d has a large intersection. Our results characterize conditions that are sufficient for the intersection of a family of convex sets to contain a "witness set" which is large under some concave or log-concave measure. The possible witness sets include ellipsoids, zonotopes, and H-convex sets. Our results also bound the complexity of finding the best approximation of a family of convex sets by a single zonotope or by a single H-convex set. We obtain colorful and fractional variants of all our Helly-type theorems.

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Tverberg's theorem bounds the number of points R d needed for the existence of a partition into r parts whose convex hulls intersect. If the points are colored with N colors, we seek partitions where each part has at most one point of each color. In this manuscript, we bound the number of color classes needed for the existence of partitions where the convex hulls of the parts intersect even after any set of t colors is removed. We prove asymptotically optimal bounds for t when r ≤ d + 1, improve known bounds when r > d + 1, and give a geometric characterization for the configurations of points for which t = N − o(N ).

In this paper, we study a family of take-away games called α-tag, parametrized by a real number α ≥ 1. We show that for any given α, there is a half-open interval I α containing α such that the set of losing positions for α-tag is the same as the set of losing positions for β-tag if and only if β ∈ I α . We then end with some results and conjectures on the nature of these intervals.