We prove polarity duality for covering problems in Hilbert geometry. Let $G$ and $K$ be convex bodies in $\mathbb{R}^d$ where $G \subset \operatorname{int}(K)$ and $\operatorname{int}(G)$ contains the origin. Let $N^H_K(G,α)$ and $S^H_K(G,α)$ denote, respectively, the minimum numbers of radius-$α$ Hilbert balls in the geometry induced by $K$ needed to cover $G$ and $\partial G$. Our main result is a Hilbert-geometric analogue of the König-Milman covering duality: there exists an absolute constant $c \geq 1$ such that for any $α\in (0,1]$, \[ c^{-d}\,N^H_{G^{\circ}}(K^{\circ},α) ~ \leq ~ N^H_K(G,α) ~ \leq ~ c^{d}\,N^H_{G^{\circ}}(K^{\circ},α), \] and likewise, \[ c^{-d}\,S^H_{G^{\circ}}(K^{\circ},α) ~ \leq ~ S^H_K(G,α) ~ \leq ~ c^{d}\,S^H_{G^{\circ}}(K^{\circ},α). \] We also recover the classical volumetric duality for translative coverings of centered convex bodies, and obtain a new boundary-covering duality in that setting. The Hilbert setting is subtler than the translative one because the metric is not translation invariant, and the local Finsler unit ball depends on the base point. The proof involves several ideas, including $α$-expansions, a stability lemma that controls the interaction between polarity and expansion, and, in the boundary case, a localized relative isoperimetric argument combined with Holmes--Thompson area estimates. In addition, we provide an alternative proof of Faifman's polarity bounds for Holmes--Thompson volume and area in the Funk and Hilbert geometries.

翻译：我们证明了希尔伯特几何中覆盖问题的极性对偶性。设 $G$ 和 $K$ 是 $\mathbb{R}^d$ 中的凸体，满足 $G \subset \operatorname{int}(K)$ 且 $\operatorname{int}(G)$ 包含原点。令 $N^H_K(G,α)$ 和 $S^H_K(G,α)$ 分别表示在由 $K$ 诱导的几何中，覆盖 $G$ 和 $\partial G$ 所需半径为 $α$ 的希尔伯特球的最小数量。我们的主要结果是 König-Milman 覆盖对偶性的希尔伯特几何类比：存在绝对常数 $c \geq 1$，使得对任意 $α\in (0,1]$，有 \[ c^{-d}\,N^H_{G^{\circ}}(K^{\circ},α) ~ \leq ~ N^H_K(G,α) ~ \leq ~ c^{d}\,N^H_{G^{\circ}}(K^{\circ},α), \] 且类似地， \[ c^{-d}\,S^H_{G^{\circ}}(K^{\circ},α) ~ \leq ~ S^H_K(G,α) ~ \leq ~ c^{d}\,S^H_{G^{\circ}}(K^{\circ},α). \] 我们还恢复了中心凸体平移覆盖的经典体积对偶性，并在此设定下获得了新的边界覆盖对偶性。希尔伯特设定比平移设定更为精细，因为该度量不是平移不变的，且局部 Finsler 单位球依赖于基点。证明涉及多个思想，包括 $α$-扩张、控制极性与扩张之间相互作用的稳定性引理，以及在边界情形下结合 Holmes--Thompson 面积估计的局部相对等周论证。此外，我们给出了 Faifman 关于 Funk 和希尔伯特几何中 Holmes--Thompson 体积与面积极性界的另一种证明。