论文标题
定量Steinitz定理：多项式结合
Quantitative Steinitz Theorem: A polynomial bound
论文作者
论文摘要
经典的Steinitz Theorem指出，如果原点属于集合$ s \ subset \ mathbb {r}^d $的凸壳的内部，则最多有$ s $的$ 2D $点的凸壳包含室内的原点。 Bárány，Katchalski和Pach证明了Steinitz定理的以下定量版本。令$ q $为$ \ mathbb {r}^d $中的凸polytope，其中包含标准的欧几里得单元球$ \ mathbf {b}^d $。然后，有$ q $的最多$ 2D $顶点，其凸壳$ q^\ prime $满足\ [r \ mathbf {b}^d \ subset q^\ prime \ prime \]，并带有$ r \ geq d^{ - 2d} $。他们猜想$ r \ geq c d^{ - 1/2} $具有通用常数$ c> 0 $。我们证明$ r \ geq \ frac {1} {5d^2} $，是$ r $的第一个多项式下限。此外，我们表明$ r $不大于$ \ frac {2} {\ sqrt {d}} $。
The classical Steinitz theorem states that if the origin belongs to the interior of the convex hull of a set $S \subset \mathbb{R}^d$, then there are at most $2d$ points of $S$ whose convex hull contains the origin in the interior. Bárány, Katchalski, and Pach proved the following quantitative version of Steinitz's theorem. Let $Q$ be a convex polytope in $\mathbb{R}^d$ containing the standard Euclidean unit ball $\mathbf{B}^d$. Then there exist at most $2d$ vertices of $Q$ whose convex hull $Q^\prime$ satisfies \[ r \mathbf{B}^d \subset Q^\prime \] with $r\geq d^{-2d}$. They conjectured that $r\geq c d^{-1/2}$ holds with a universal constant $c>0$. We prove $r \geq \frac{1}{5d^2}$, the first polynomial lower bound on $r$. Furthermore, we show that $r$ is not be greater than $\frac{2}{\sqrt{d}}$.
