Unlinking theorem for symmetric convex functions

The three-dimensional Gaussian product inequality

“…For example, the Gaussian correlation inequality recently proved by Royen [16] (cf. Lata la and Matlak [11]) plays an important role in small ball probabilities (Li [12], Shao [17]) and the U-conjecture (Kagan, Linnik and Rao [9], Bhandari and DasGupta [5], Hargé [8], Bhandaria and Basu [4]). Another famous inequality associated with Gaussian distributions is the Gaussian product conjecture, which is still an open problem.…”

Characterization of equality in the correlation inequality for convex functions, the U-conjecture

“…We begin to prove some elementary facts concerning convex functions. The following reasoning process is inspired by the one of Bhandari and DasGupta [6]. Lemma 3.1.…”

“…Lemma 3.1. If ϕ : R → R is a convex and non-constant function then lim t→+∞ ϕ(t) = +∞ or lim t→−∞ ϕ(t) = +∞ [6].…”

“…anihpb.2004.05.005 have shown that if P and Q are two convex, positive polynomials independent with respect to µ n and if P (0) = 0 then P and Q are unlinked. Related to this problem, Bhandari and DasGupta [6] proved in 1994 that two convex and even functions f and g are unlinked if they are uncorrelated (that is if fg dµ n = f dµ n g dµ n ) under an additional hypothesis. This hypothesis is related to the Gaussian correlation conjecture which is still a conjecture in dimension greater than two (see [7] for further details and references on this conjecture), so, the result of Bhandari and DasGupta is not proved in the general case, that is, without this additional hypothesis.…”

On the unlinking conjecture of independent polynomial functions

“…However, the conjecture in its generality appears to be an extremely difficult problem to handle. Bhandari and DasGupta [1] proved that if P and Q are both symmetric convex functions, not necessarily polynomials, then the independence of P and Q implies P and Q are unlinkable. However, their proof was itself based on the correlation inequality P (A ∩ B) P (A)P (B), where both A and B are convex symmetric sets in R n .…”