Inside s-inner product sets and Euclidean designs (Q2428639)

Let \(X\) be finite set in the Euclidean space \(R^d\); for \(x, y \in R^d\), denote their standard inner product by \((x,y)\). We say \(X\) is an \(s\)-inner product set if the set of the usual inner products of any two distinct points in \(X\) has size \(s\). The author provides a special upper bound for the cardinality of an \(s\)-inner product set on concentric spheres. This upper bound coincides with the known lower bound for the size of an Euclidean \(s\)-design. The author also proves the non-existence of 2- or 3-inner product sets on two concentric spheres attaining the upper bound for any \(d >1\). The efficient property needed to prove the upper bound for an \(s\)-inner product set gives a new concept called inside \(s\)-inner product sets: Let \(B(x):=\{(x,y)| y \in X, x \neq y, \| x\| \geq \| y \|\}\); if \(| B(x)| \leq s\) for each \(x\in X\) then \(X\) is called an inside \(s\)-inner product set. The author characterizes the most known tight Euclidean designs as inside \(s\)-inner product sets attaining the upper bound.