Let R2 be the usual two-dimensional plane with the Eucledean norm 1 . I. By CONV we denote the set of all convex compact subsets of R2. The Hausdorff distance between two elements A,, A, of CONV is given by h(A,,A,)=inf{t>O: A,cA,+fB, A,cA,+fB}, where B=(PER2: IPI 3 we denote by POLY, the set of all convex polygons with not more than n vertices. The elements of POLY, will be called n-gons. The ngon A, is said to be a best Hausdorff approximation in POLY, for the set A E CONV if inf{h(A, A): A E POLY,} = h(A, A,). The existence of at least one best Hausdorff approximation for any A E CONV follows from the wellknown Blaschke “selection theorem” asserting that every bounded sequence of n-gons (n fixed) contains a subsequence converging in the Hausdorff metric to some n-gon. In general, as examples like the unit circle or the unit square show, the best approximation is not unique. Nevertheless the “majority” of the elements of CONV have unique best approximation in any POLY,, n > 3. The “majority” here means: with an exception of some first Baire category subset of the locally compact metric space (CONV, h), all convex compact subsets of R2 have unique best approximation in POLY, for every n > 3 (Theorem 3.5). To prove this we give (and use) a necessary condition for A E POLY, to be a best approximation for A E CONV. This condition (Theorem 2.1) coincides with the classical alternating condition in