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Let P be a set of n points in R d . It was conjectured by Schur that the maximum number of (d − 1)-dimensional regular simplices of edge length diam(P ), whose every vertex belongs to P , is n. We prove this statement under the condition that any two of the simplices share at least d − 2 vertices and we conjecture that this condition is always satisfied.

Let P be a set of n points in R d . It was conjectured by Schur that the maximum number of (d − 1)-dimensional regular simplices of edge length diam(P ), whose every vertex belongs to P , is n. We prove this statement under the condition that any two of the simplices share at least d − 2 vertices and we conjecture that this condition is always satisfied.

By a polygonization of a finite point set S in the plane we understand a simple polygon having S as the set of its vertices. Let B and R be sets of blue and red points, respectively, in the plane such that B ∪ R is in general position, and the convex hull of B contains k interior blue points and l interior red points. Hurtado et al. found sufficient conditions for the existence of a blue polygonization that encloses all red points. We consider the dual question of the existence of a blue polygonization that excludes all red points R. We show that there is a minimal number K = K(l), which is polynomial in l, such that one can always find a blue polygonization excluding all red points, whenever k ≥ K. Some other related problems are also considered.