Abstract: I'll discuss some problems of geometric optimization that began with an attempt to understand the means of chords on a closed planar curve of fixed length. By a chord we refer to the length of the line segment joining two points on the curve, differing by arclength \alpha. One can consider different means of this quantity, for example L^p means with respect to arclength, or with respect to a weight proportional to curvature. For example, for 1 < p < 2, in the unweighted case the means of chords are shown to be maximized by the circle. The situation is different for sufficiently high p and for the weighted means we consider, and we can identify some cases of optimum while the situation is open in other cases. In the weighted case we can impose the constraint of convexity and identify a wider family of convex functionals for which the maximizing shapes are triangles or segments. Among other things, the more general problems include maximizing Hausdorff distances between sets, under some geometric constraints.