2018/09/05 by Dobbins, Michael Gene, Frick, Florian
#51M04 #51M20 #52B11 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1809.01613
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. We show that we can increase the dimension of P 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.