AI 中文总结
该研究针对希尔伯特几何中的凸化装填问题，证明了无维数的原-极界，建立了原凸化装填数与反向极体普通装填数的多项式关系，为将赋范空间的相关结果推广到希尔伯特几何提供了关键支撑。
AI 中文摘要
设$G$和$K$是$\u211d^d$中的凸体，满足$0 \u2208 \text{int} G$且$G \text{int} K$。给定$\u03b1>0$，希尔伯特装填数$M_H(G, K; \u03b1)$定义为$G$中满足任意两点在由$K$定义的希尔伯特几何下的距离至少为$\u03b1$的点集的最大基数。希尔伯特凸化装填数$\u0302{M}_H(G, K; \u03b1)$定义为$G$中满足每个点与其前面所有点的凸包的距离至少为$\u03b1$的点序列的最大长度。我们证明了希尔伯特几何中凸化装填的无维数原-极界。设$G^\u25e6$和$K^\u25e6$分别表示极体，我们证明存在绝对常数$C, c > 0$，使得对任意$\u03b1>0$，有\u0302{M}_H(G, K; \u03b1) \u2264 C \u0302{M}_H(K^\u25e6, G^\u25e6; c\u03b1)^2 M_H(K^\u25e6, G^\u25e6; c\u03b1)。作为直接推论，我们有\u0302{M}_H(G, K; \u03b1) \u2264 C M_H(K^\u25e6, G^\u25e6; c\u03b1)^3。因此，原凸化装填数可以由反向极体的普通装填数的固定次多项式界定，其中绝对常数与维数无关。该研究受装填与覆盖数的对偶猜想启发，该猜想将用$K$覆盖$G$与用$G^\u25e6$覆盖$K^\u25e6$联系起来。我们的结果是将Artstein、Milman、Szarek和Tomczak-Jaegermann的工作从赋范空间推广到希尔伯特几何的关键一步。
英文摘要
Let $G$ and $K$ be convex bodies in $\mathbb{R}^d$, where $0 \in \text{int} G$ and $G \subset \text{int} K$. Given $α> 0$, the Hilbert packing number $M_H(G, K; α)$ is the maximum cardinality of a set of points in $G$, each pair of which is separated by a distance of at least $α$ in the Hilbert geometry defined by $K$. The Hilbert convexified packing number $\widehat{M}_H(G, K; α)$ is the maximum length of a sequence of points in $G$, such that each point is separated by distance at least $α$ from the convex hull of its predecessors. We prove a dimension-free primal-polar bound for convexified packing in Hilbert geometry. Letting $G^\circ$ and $K^\circ$ denote the polar bodies, we show that there exist absolute constants $C, c > 0$ such that, for every $α> 0$, \[ \widehat{M}_H(G, K; α) ~ \leq ~ C \cdot \widehat{M}_H(K^\circ, G^\circ; cα)^2 M_H(K^\circ, G^\circ; cα). \] As a direct corollary, we have \[ \widehat{M}_H(G, K; α) ~ \leq ~ C \cdot M_H(K^\circ, G^\circ; cα)^3. \] Thus, the primal convexified packing number is bounded by a fixed polynomial in the ordinary packing number for the reversed polar bodies, with absolute constants independent of the dimension. This is motivated by the duality conjecture for packing and covering numbers, which relates covering $G$ by $K$ to covering $K^\circ$ by $G^\circ$. Our results represent a key step in extending the work of Artstein, Milman, Szarek, and Tomczak-Jaegermann from normed spaces to Hilbert geometries.
