The 123 Theorem and its extensions and Raphael Yuster Summary: The 123 Theorem and its extensions Noga Alon and Raphael Yuster Department of Mathematics Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv, Israel Abstract It is shown that for every b > a > 0 and for every two independent identically distributed real random variables X and Y Prob[|X - Y | b] < (2 b/a - 1)Prob[|X - Y | a]. This is tight for all admissible pairs a, b. Higher dimensional extensions are also considered. 1 Introduction Our first result in this note is the following theorem, which we name after the three constants in its statement. Theorem 1.1 (The 123 Theorem) Let X and Y be two independent, identically distributed real random variables. Then Prob[|X - Y | 2] < 3Prob[|X - Y | 1]. The problem of determining the smallest possible constant C so that for every two independent, identically distributed (=i.i.d.) real random variables the inequality Prob[|X - Y | 2] CProb[|X - Y | 1] Collections: Mathematics