Let $X_1, \cdots, X_N$ be i.i.d. $d$-dimensional random vectors, the exact distribution of $X$ is not very important in my application (as long as it continuous) so pick the one that works best, but ideally multivariate normal. Then let $Y$ be another $d$-dimensional random vector from a distribution that share the same central moments as the first distribution (i.e. the first distribution with shifted mean). Could one then find:

$$E\left(\min_{\alpha_1,\cdots,\alpha_N}\left\|Y-\sum_{i=1}^N\alpha_i X_i\right\|\right)\quad s.t. \sum_{i=1}^N \alpha_i = 1 \wedge \alpha_i\geq 0$$

or maybe bounds on it? Or maybe the probability $P\left(\min_{\alpha_1,\cdots,\alpha_N}\left\|Y-\sum_{i=1}^N\alpha_i X_i\right\|=0\right)$?