Open Collections

Description

For any separable Banach space $(F,\| \cdot \|)$ and independent $F$-valued random vectors $X$ and $Y$ such that ${\mathbf E} \|X\|, {\mathbf E} \|Y\|< \infty$, we have \[ \inf_{z \in F} ({\mathbf E}\|X-z\|+{\mathbf E}\|Y-z\|) \leq 3 \cdot {\mathbf E}\| X-Y\|. \] Indeed, it suffices to consider $z=({\mathbf E} X+{\mathbf E} Y)/2$ and use Jensen's inequality. Assaf Naor asked whether the constant $3$ in the inequality is optimal. We will discuss this and related problems.