Let $V$ be a set of $n$-dimensional vectors such that, for each ${\bf v}\in V$ and for each index $i\in [n-1]$, we have $0\le v_{i+1}\le v_i$. Let $P(\cdot)$ be a discrete probability distribution over $V$.

We are given the expected number $\mu$ of positive (i.e. non-null) vector components of a vector selected from $V$ with $P(\cdot)$.

**Question**: What is the *maximum* value of $\alpha$ for which the following inequality always holds?
$$\max_{{\bf u}\in V} \sum_{i=1}^{\lceil \mu \rceil} u_i \ge \alpha\,\mathbb{E}_{P({\bf v})} ||{\bf v}||_1$$

(Does it always hold even for $\alpha=1$?)

maximumvalue of $\alpha$? $\endgroup$ – Iosif Pinelis May 3 at 14:36