Inside S-inner product sets and Euclidean designs

Let X be a finite subset of Euclidean space R. We define for each x ∈ X, B(x) := {(x, y) | y ∈ X, x 6= y, (x, x) ≥ (y, y)} where (, ) denotes the standard inner product. X is called an inside s-inner product set if |B(x)| ≤ s for all x ∈ X. In this paper, we prove that the cardinalities of inside s-inner product sets have the Fisher type upper bound. An inside s-inner product set is said to be tight if its cardinality attains the Fisher type upper bound. Tight inside s-inner product sets are closely related to tight Euclidean designs. Indeed, many tight Euclidean designs are tight inside s-inner product sets. Generally, a tight inside s-inner product set becomes a Euclidean s-design with a weight function which may not be positive. In the last part of this paper, we prove the non-existence of tight 2or 3-inner product sets supported by a union of two concentric spheres. In order to prove this result, we give a new upper bound for the finite subset of R which does not have positive inner products.