We prove that the moduli of U-convexity, introduced by Gao (1995), of the ultrapower X̃ of a Banach space X and of X itself coincide whenever X is super-reflexive. As a consequence, some known results have been proved and improved. More precisely, we prove that uX(1) > 0 implies that both X and the dual space X∗ of X have uniform normal structure and hence the “worth” property in Corollary 7 of ...

Let α ≥ 1 and let (X, d, μ) be an α-homogeneous metric measure space with conformal Assouad dimension equal to α. Then there exists a weak tangent of (X, d, μ) with uniformly big 1-modulus.

Let T (x; r) denote the total occupation measure of the ball of radius r centered at x for Brownian motion in IR 3. We prove that sup jxjj1 T (x; r)=(r 2 jlog rj) ! 16== 2 a.s. as r ! 0, thus solving a problem posed by Taylor in 1974. Furthermore, for any a 2 (0; 16== 2), the Hausdorr dimension of the set of \thick points" x for which limsup r!0 T (x; r)=(r 2 jlog rj) = a, is almost surely 2 ? ...