International Congress of Mathematicians

816 R. FatalaIt is not hard to see that Theorem 3.1 implies the isoperimetric inequality(2.1). Indeed we have for any Borei set A in R"Q-H-YniAt)) = #- 1 ( 7 „(A(A- 1 A) + (1 - A)((l - A)" 1 *^")))> A#- 1 ( 7 „(A- 1 A)) + (1 - A)#- 1 ( 7 „((l - A)- 1^")) A^T ^(^(A)) + t.Conjecture 3.1 Inequality (3.2) holds for any Borei sets in F.Ehrhard's symmetrization procedure enables us to reduce Conjecture 3.1 tothe case F = R and p = 71. We may also assume that A and B are finite unions ofintervals. At the moment the conjecture is known to hold when A is a union of atmost 3 intervals.Ehrhard's inequality has the following Prekopa-Leindler type functional version.Suppose that A G (0,1) and f,g,h:R n —¥ [0,1] are such thatthenV œ ,, GRn ^(MAx + (1 - X)yj) > A#- 1 (/(x)) + (1 - A)*- 1^))* _1 (/ hchJiXQ-HÏ /d 7 „) + (l^A)#- 1 (/' gd ln ). (3.3)We use here the convention $ _1 (0) = —oo,^1(Y)= 00 and ^00 + 00 = ^00. Atthe moment the above functional inequality is known to hold under the additionalassumption that at least one of the functions $ _1 (/),$ _1 ( I hd-/ n + t})0valid for all Lipschitz functions h : R" —¥ R with the Lipschitz seminorm ||/I||Lì P =sup{\h(x) — h(y)\ : x,y £ W 1 } < 1. However the logarithmic Sobolev inequalitydoesnot imply the isoperimetric inequality.