AI 中文总结
本文证明了最优颜色数$2d$的彩色定量Helly定理，并推广了彩色定量Steinitz定理，通过正算子归一化和彩虹基提升方法，给出了体积下界$d^{-O(d^2)}$。
AI 中文摘要
我们证明了一个关于体积的彩色定量Helly定理，其颜色数达到最优的$2d$。若$\reals^d$中$2d$个有限凸集族的每个彩虹交集体积至少为1，则其中某个族的交集体积至少为$d^{-O(d^2)}$。我们还证明了一个关于不同形状的中心在原点的椭球的彩色定量Steinitz定理。证明使用了正算子的公共归一化以及一个产生两个具有大行列式的彩虹基的提升。
英文摘要
We prove a colorful quantitative Helly theorem for volume with the optimal number $2d$ of colors. If every rainbow intersection from $2d$ finite families of convex sets in $\R^d$ has volume at least one, then the intersection of one family has volume at least $d^{-O(d^2)}$. We also prove a colorful quantitative Steinitz theorem for origin-centered ellipsoids of different shapes. The proof uses a common normalization of positive operators and a lift that produces two rainbow bases with large determinants.
