We prove several exact quantitative versions of Helly's and Tverberg's theorems, which guarantee that a finite family of convex sets in ${\mathbb{R}}^d$ 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 show that several new optimization problems can be solved with algorithms for LP-type problems. We obtain colorful and fractional variants of all our Helly-type theorems.

中文翻译：

---

凹函数的定量组合几何

我们证明了Helly和Tverberg定理的几个精确的定量版本，这保证了凸集的有限族 ${\mathbb{[R}}^d$交叉路口很大 我们的结果表征了足以使凸集族的交集包含“见证集”的条件，该“见证集”在某些凹入或凹入-凹入量度下较大。可能的见证集包括椭圆体，环带和H-凸集。我们的结果表明，使用LP类型问题的算法可以解决几个新的优化问题。我们获得所有Helly型定理的彩色和分数变体。