  The Komlos conjecture in discrepancy theory states that for some constant K
and for any m by n matrix A whose columns lie in the unit ball there exists a
+/- 1 vector x such that the infinity norm of Ax is bounded above by K. This
conjecture also implies the Beck-Fiala conjecture on the discrepancy of bounded
degree hypergraphs. Here we prove a natural relaxation of the Komlos
conjecture: if the columns of A are assigned unit real vectors rather than +/-
1 then the Komlos conjecture holds with K=1. Our result rules out the
possibility of a counterexample to the conjecture based on semidefinite
programming. It also opens the way to proving tighter efficient
(polynomial-time computable) upper bounds for the conjecture using semidefinite
programming techniques.
