Given TV points X\, x 2 , ... , XN on a unit sphere S in Euclidean d space (d > 3), we investigate the α-sum ]ζ \x -x } \ a , a > 1 -d, of their distances from a variable point x on S. We obtain an essentially best possible lower bound for the Z^-norm of its deviation from the mean value. As an application, we prove similar bounds for the α-sums Σ\ X J ~ χ k\ a of mutual distances.
Given TV points x\, ... , x N on the unit sphere S in Euclidean d space (d > 3), lower bounds for the deviation of the sum ]Γ \x-Xj\ a , a > 1 -d x £ S, from its mean value were established in terms of L λ -norms in the first part of this paper. In the present part it is shown that these bounds are best possible. Our main tool is a multidimensional quadrature formula with equal weights.
Let 5 be the surface of the unit sphere in three-dimensional euclidean space, and let a>n = X2,...,XN) be an N-tuple of points on 5. We consider the product of mutual distances p(a>f/) = Ylj^k \ x j ~ x k I a n d. f°r t ' l e variable point x on S, the product of distances p(x, (ON) = n^Li I* -Xj\ from x to the points of « # . We obtain essentially best possible bounds for p{cofi) and for m i n^ max xe sp(x, wjv)-
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