BROWNIAN MARTINGALES AND CONJECTURE OF SA KAI

For any real p greater than or equal to 0 such that the function x --> \\x\\(p) is subharmonic in R(d), M. Sakai introduced in 1988 the cons tant c(d)(p) optimal in a given isovolumetric inequality: c(d)(p) tell s how large the value at the origin of the least harmonic majorant of \\x\\(p) Can be on a domain of R(d) of given volume. We use probabilis tic tools to study this constant. We recover most of M. Sakai's result s, proving them often in a simpler and more enlightening way. We obtai n new estimates of c(d)(p) in the cases M. Sakai left open. We thoroug hly study the case d = 2, p = 4 and we prove that c(2)(4) < 1.80.