Systems of vectors in Euclidean space and an extremal problem for polynomials

Abstract

n 112 _,x ll< (2) We use induction on n. For n = 1 our assertion is obvious. We will show that if it holds for all values less than n, then it holds for n. Suppose ~ = {xl,...,xn} is an arbitrary L-system. For i = i, . . ., n we put A i = { ] ~ { t . . . . . n}: ] • i , ( x ~ , x j ) ~ 0 } . (3 ) We c o n s i d e r two c a s e s . The f i r s t c a s e : f o r e a c h i = 1 , . . . , n t h e c a r d i n a l i t y o f A i i s a t m o s t n 2 / 3 . T h e n n 2 n I ] ~ I Xi[t = 2 i=1 (Zt, Xji ) = ~ = i ( xi' Xi) + ~'i~j( xi' Xj)= = n J r ~ . # j , (xi, xp§ (xt, xr ---n + ~ = 1 2 J E A i (xi' Xj) = n n =n + 21=1(xi, ~,j~AiXj)<n + y,i=xil2i~AxjH. (4)