We define a geometric condition which is sufficient for the Diophantine extremality of a measure on Euclidean n-space, thus generalizing the notion of "friendliness" considered by Kleinbock, Lindenstrauss, and Weiss ('04). This condition is very general and is satisfied by many classes of dynamically defined measures, including all exact dimensional measures of sufficiently large dimension, Patterson-Sullivan measures of nonplanar geometrically finite groups, and conformal measures of infinite IFSes. We will also give some examples of non-extremal measures coming from dynamics, illustrating where the theory must halt.

These results are joint-work with Tushar Das, Lior Fishman and David Simmons.