Let $(X, d)$ be a metric space and $C \subseteq 2^X$ -- a collection of special objects. In the $(X,d,C)$-chasing problem, an online player receives a sequence of online requests $\{B_t\}_{t=1}^T \subseteq C$ 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 $(X,d,C)$-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，$C \subseteq 2^X$为特定对象的集合。在$(X,d,C)$-追踪问题中，在线玩家接收到一系列在线请求$\{B_t\}_{t=1}^T \subseteq C$，并做出轨迹响应$\{x_t\}_{t=1}^T$使得$x_t \in B_t$。该响应产生移动成本$\sum_{t=1}^T d(x_t, x_{t-1})$，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成本与事后最优移动成本之间的最坏情况比率。在此框架下，若存在具有有限竞争比的在线算法，则称$(X,d,C)$-追踪问题为$\textit{可追踪的}$。对于实赋范向量空间上的凸体追踪问题（CBC），(Bubeck et al. 2019) 证明了该问题的可追踪性。进一步，在向量空间设定中，环境空间的维度似乎成为控制竞争比大小的关键因素。事实上，(Sellke 2020) 近期给出了在任意实赋范向量空间$(\mathbb{R}^d, ||\cdot||)$上具有$d$竞争比的在线算法，而我们将简要展示获得相同设定下CBC问题$\Omega(d^c), \enspace c > 0$形式新颖下界的一般策略。本文同时证明，度量空间的$\textit{倍测度维度}$和$\textit{Assouad维度}$对该度量空间上的球体追踪难度不具备控制作用。具体而言，我们证明对于任意足够大的$\rho \in \mathbb{R}$，存在一个倍测度维度为$\Theta(\rho)$且Assouad维度为$\rho$的度量空间$(X,d)$，使得在该空间的一般球体追踪设定中，没有任何在线选择器能够实现有限竞争比。