Sharp small-deviation inequalities for sums of independent nonnegative random variables

Let $(X_1,\dots ,X_n)$ be independent nonnegative random variables with $\mathbb{E}{X}_i\le 1$, and write $S={\sum }_i{X}_i$. For $\delta >0$, we prove that $$\mathbb{P}(S<\mathbb{E}S+\delta )\ge b_{n,\delta },$$ where $b_{n,\delta }=\delta (n\mathrm{/}(n+\delta ){)}^n$ for $0<\delta <1$ and $b_{n,\delta }=(1-1\mathrm{/}(n+\delta ){)}^n$ for $\delta \ge 1$. The bound is sharp for every $n$ and $\delta \ge 1$. In particular, since $b_{n,\delta }\ge e^{-1}$ for $\delta \ge 1$, our result proves Feige’s conjecture [Feige, 2004] in the affirmative for $\delta \ge 1$. The proof is found by ChatGPT 5.6 Pro. It combines the exact Dirichlet calibration theorem of Vlassis and Thomas [Vlassis and Thomas, 2026], which resolves Gaffke’s conjecture in statistics, with results in convex geometry including Gr"unbaum’s centroid theorem [Gr"unbaum, 1960] and its generalization by Letwin and Yaskin [Letwin and Yaskin, 2024].