Let $\emptyset\neq M\subseteq \mathbb{R}^n$ be a compact set, $X:\Omega\rightarrow M$ be a random vector defined on a complete probability space $(\Omega,\mathcal{F},\mathbb{P})$ and suppose that $\mu:=E[X]\in \mathbb{R}^n$ exists; i.e. $X$ is Bochner integrable. Suppose also that the metric entropy of $M$, denoted by $H_{\epsilon}(M)$ (intuitively $<<n$), and defined by $\log_2(N_{\epsilon}(M))$ where $N_{\epsilon}(M)$ is the minimum number of balls in $M$ of radius $\epsilon$ required to cover $M$.

Are there any known "good" bounds the variance of $X$ given this data?

In the univariate case, I came across Popoviciu's inequality but this only seems to work for $n=1$ and it does not incorporate the information we know on $M$.

Thanks in advance!


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