Barycenters of points in polytope skeleta

The first author showed that for a given point p in an nk-polytope P there are n points in the k-faces of P, whose barycenter is p. Herewecompletely classify n-tuples of dimensions (k₁,…,k_n)thatsumtonk such that there are n points from faces of these prescribed dimensions whose barycenter is p. WeshowthatwecanincreasethedimensionofP by r, if we allow r of the points to be in (k + 1)-faces. While we can force points with a prescribed barycenter into faces of dimensions k and k+1, we show that the gap in dimensions of these faces can never exceed one. We also investigate the weighted analogue of this question, where a convex combination with predetermined coefficients of n points in k-faces of an nk-polytope is supposed to equal a given target point. While weights that are not all equal may be prescribed for certain values of n and k, any coefficient vector that yields a point different from the barycenter cannot be prescribed for fixed n and sufficiently large k.