Let $(X, d)$ be a metric space and $\mathcal{C} \subseteq 2^X$ -- a collection of special objects. In the $(X,d,\mathcal{C})$-chasing problem, an online player receives a sequence of online requests $\{B_t\}_{t=1}^T \subseteq \mathcal{C}$ and responds with a trajectory $\{x_t\}_{t=1}^T$ such that $x_t \in B_t$. This response incurs a movement cost $\sum_{t=1}^T d(x_t, x_{t-1})$, and the online player strives to minimize the competitive ratio -- the worst case ratio over all input sequences between the online movement cost and the optimal movement cost in hindsight. Under this setup, we call the $(X,d,\mathcal{C})$-chasing problem $\textit{chaseable}$ if there exists an online algorithm with finite competitive ratio. In the case of Convex Body Chasing (CBC) over real normed vector spaces, (Bubeck et al. 2019) proved the chaseability of the problem. Furthermore, in the vector space setting, the dimension of the ambient space appears to be the factor controlling the size of the competitive ratio. Indeed, recently, (Sellke 2020) provided a $d-$competitive online algorithm over arbitrary real normed vector spaces $(\mathbb{R}^d, ||\cdot||)$, and we will shortly present a general strategy for obtaining novel lower bounds of the form $\Omega(d^c), \enspace c > 0$, for CBC in the same setting. In this paper, we also prove that the $\textit{doubling}$ and $\textit{Assouad}$ dimensions of a metric space exert no control on the hardness of ball chasing over the said metric space. More specifically, we show that for any large enough $\rho \in \mathbb{R}$, there exists a metric space $(X,d)$ of doubling dimension $\Theta(\rho)$ and Assouad dimension $\rho$ such that no online selector can achieve a finite competitive ratio in the general ball chasing regime.

翻译：设 $(X, d)$ 为一个度量空间，$\mathcal{C} \subseteq 2^X$ 为一类特殊对象的集合。在 $(X,d,\mathcal{C})$-追逐问题中，在线玩家接收一系列在线请求 $\{B_t\}_{t=1}^T \subseteq \mathcal{C}$，并回应以轨迹 $\{x_t\}_{t=1}^T$，使得 $x_t \in B_t$。该回应产生移动代价 $\sum_{t=1}^T d(x_t, x_{t-1})$，在线玩家致力于最小化竞争比——即所有输入序列下在线移动代价与事后最优移动代价之间的最坏情况比值。在此设定下，若存在具有有限竞争比的在线算法，则称 $(X,d,\mathcal{C})$-追逐问题为可追逐的。对于实赋范向量空间上的凸体追逐问题（CBC），Bubeck 等人（2019）证明了该问题的可追逐性。此外，在向量空间设定中，环境空间的维度似乎是控制竞争比大小的因素。事实上，近期 Sellke（2020）在任意实赋范向量空间 $(\mathbb{R}^d, ||\cdot||)$ 上提出了一个 $d$-竞争的在线算法，其后我们将简要介绍在同一设定下针对 CBC 问题获得 $\Omega(d^c), \enspace c > 0$ 形式新下界的一般策略。本文还证明，度量空间的倍测度维数和 Assouad 维数对该度量空间上的球体追逐问题的难度没有控制作用。具体而言，我们证明：对于任意足够大的 $\rho \in \mathbb{R}$，存在一个倍测度维数为 $\Theta(\rho)$ 且 Assouad 维数为 $\rho$ 的度量空间 $(X,d)$，使得在一般球体追逐机制下，没有在线选择器能实现有限竞争比。