  We derive some new results on the k-th barycentric Olson constants of abelian
groups (mainly cyclic). This quantity, for a finite abelian (additive) group
(G,+), is defined as the smallest integer l such that each subset A of G with
at least l elements contains a subset with k elements {g_1, ..., g_k}
satisfying g_1 + ... + g_k = k g_j for some 1 <= j <= k.
