# Ensuring a minimum dot-product

Consider a unit spherical surface in $N$ dimensions. I have a set of $m$ vectors $\{\mathbf{v_{i}}\}$ lying on this surface. What is the minimum value of $m$ required such that for every vector $\mathbf{x}$ lying on the spherical surface, there exists at least one $\mathbf{v_{i}}$ such that $\vert \mathbf{x.v_{i}} \vert \geq \lambda$ (some threshold)?

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It seems like you might need some conditions on the set of vectors. –  AnonymousCoward Jul 25 '12 at 6:57
This is a sphere covering problem on the surface of the sphere. It should be fairly easy to estimate the order of magnitude of $m$, but the precise value as a function of $\lambda$ sounds intractable. –  Erick Wong Jul 25 '12 at 7:00
@ErickWong: I was thinking of this as a problem of obtaining the solid angle in $N$ as a function of $\cos^{-1}(\lambda)$. Is this the approximation you are referring to? –  kashyap1123 Jul 25 '12 at 8:17