# Order Statistics and High Dimension Geometry

Suppose that I have iid random variables $$\mathrm U_n \sim \mathrm U(0,1)$$. Then, for $$\mathrm Y_m$$ defined as,

$$\mathrm Y_m = \min_{n \in [1,m]} \mathrm U_n$$

it is easy to compute $$\mathbb{E}[\mathrm Y_m] = \frac1{m+1}$$.

Now consider a compact convex region $$\mathcal{C}$$ in higher dimensions $$\mathbb{R}^k$$ and let $$\mathbf p_n$$ be fixed points in $$\mathcal C$$ where $$\mathbf p_n$$ may lie on the boundary of $$\mathcal C$$. Let $$\mathrm D_n$$ denote iid random variables in $$\mathcal C$$ such that $$\mathbb P_{\mathrm D_n}(S) \geq \kappa \frac{\mathrm{vol}(S)}{\mathrm{vol}(\mathcal C)}~\forall S\subseteq \mathcal C$$ for some
fixed $$\kappa < 1$$. Thus $$\mathrm D_n$$ denotes a "skewed" uniform distribution of points in $$\mathcal C$$.

Suppose that $$\mathrm U_n$$ now corresponds to the distance of $$\mathrm D_n$$ from $$\mathbf p_n$$ ie. $$\mathrm U_n \triangleq ||\mathrm D_n - \mathbf p_n||$$. Can someone point me in the right direction to find upper bounds on $$\mathbb E[\mathrm Y_n]$$ in such cases ( or direct me to references ) ?