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probability-statistics / Probability

Feige's Conjecture

Let $X_1,\ldots,X_n$ be independent nonnegative random variables with $\mathbb{E}X_i \le 1$, and let $S$ be their sum. Is $\mathbb{P}(S < \mathbb{E}S + 1) \ge 1/e$? Feige proved the constant $1/13$ and conjectured the sharp $1/e$. Three independent July 2026 proofs settle it, both building on the Vlassis-Thomas calibration theorem; the sharper one determines the optimal small-deviation bound for every deviation $\delta \ge 1$.

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probability-statisticsJul 27, 2026Significance 35/100Registry: lean verified

Feige's Conjecture

Prior state unknown→proved

Let $X_1,\ldots,X_n$ be independent nonnegative random variables with $\mathbb{E}X_i \le 1$, and let $S$ be their sum. Is $\mathbb{P}(S < \mathbb{E}S + 1) \ge 1/e$? Feige proved the constant $1/13$ and conjectured the sharp $1/e$. Three independent July 2026 proofs settle it, both building on the Vlassis-Thomas calibration theorem; the sharper one determines the optimal small-deviation bound for every deviation $\…

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Let $X_1,\ldots,X_n$ be independent nonnegative random variables with $\mathbb{E}X_i \le 1$, and let $S$ be their sum. Is $\mathbb{P}(S < \mathbb{E}S + 1) \ge 1/e$? Feige proved the constant $1/13$ and conjectured the sharp $1/e$. Three independent July 2026 proofs settle it, both building on the Vlassis-Thomas calibration theorem; the sharper one determines the optimal small-deviation bound for every deviation $\delta \ge 1$.

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