# Reference: Good bounds for Variance of a Random Vector with Known Mean Supported on a Compact Set of Low Metric Entropy

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 $$<), and defined by $$\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 $$M$$.