Study on the distances and arrangement of finite-point-set

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1.Let G be the graph obtained from a complete graph with countably many vertices by removing an edge. Then G is not an integral-distance graph in any dimension, but it is a rational distance graph in the plane. If a complete graph with n vertices can be realized as an integral distance graph in the plane in such a way that no three vertices lie on a line, and no four vertices lie on a circle, then the complete n-partite graph K(a_1,a_2,【triple bond】,a_n) is a rational distance graph in the plane, where a_k=(k-1 choose 2)+(k-1 choose 3)+1.2.For any n>4, and any k>1, there is an n-point-set such that the center of the minimal star of the n-point-set in p-norm (p=1,2,【triple bond】,k) are all distinct. But for any 4-point-set the center of the minimal star are the same point for any norm. (Joint work with M.Watanabe).3.Let X be a point-set with at least m+2 points. The map from the family of m+1 point-set of X to the nonnegative reals that assigns to each (m+1)-point-set, the m-dimensional volume of the convex hull of the (m+1)-point-set, is a hemimetric and satisfies the m-dimensional simplex inequality. For each m, we can define the "bound" s(m) of m-dimensional simplex inequality. This bound s(m) determines the "configuration" X to some extent. For example, if |X|>4, then the three statements s(2)=2,s(3)=3, and [X is the vertex-set of a regular simplex] are equivalent. We calculated s(m) for regular polyhedra in 3-space. Though s(m)=3 for the n-dimensional cross-polytope, n>m-1>1, the value s(m) for n-cube tends to 1 as n tends to infinity. (Joint work with M.Deza and M.Dutour).