We will discuss best and random approximations of spindle convex sets by convex disc-polygons. Spindle convex sets in the Euclidean plane are sets of circumradius not greater than one with the property that together with any pair of its points the set contains every short circular arc of radius at least one connecting these points. A convex disc-polygon is the intersection of a finite number of closed unit radius circular discs.

In this talk, we will prove sharp estimates of the order of convergence of best approximations of planar spindle convex sets by inscribed and circumscribed convex disc-polygons with respect to the Hausdorff metric, and the measures of deviation defined by perimeter and area differences. We will also discuss some aspects of random approximations of spindle convex sets by inscribed convex disc-polygons.

The results presented in this talk are joint with Viktor Vigh.