An inequality conjectured by Hajela and Seymour arising in combinatorial geometry

Abstract

In a recent paper, D. Hajela and P. Seymour proved that for 0≦b ₁≦b ₂≦1, α=(log₂ 3)/2,

$$b_1^\alpha b_2^\alpha + (1 - b_1 )^\alpha b_2^\alpha + (1 - b_1 )^\alpha (1 - b_2 )^\alpha \leqq 1,$$

and drew from this inequality a variety of interesting results in combinatorial geometry. They also conjectured a generalization of the inequality ton variables, which they showed to imply a lower bound on the number of different sequences obtained when members ofn sets of zero-one sequences are added to one another.

We prove their conjecture, not easy to verify even for small values ofn, using complex-variable theory.