I'll discuss an approach for compressing the empirical measure in the context of finite dimensional reproducing kernel Hilbert spaces (RKHSs). In this context, an embedding of the empirical measure is contained within a convex set within an RKHS and can be approximated by using convex optimization techniques. Such an approximation gives under certain conditions rise to a small core-set of data points. A key quantity that controls the size of such a core-set is the size of the largest ball that fits within the convex set and which is centred at the embedding of the empirical measure. I will give a high-level overview of how high probability lower bounds on the size of such a ball can be derived before discussing how the approach can be adapted to standard problems like non-linear regression.  (The talk will be based on an extended version of https://arxiv.org/pdf/2204.08847.pdf)