Shapiro inequality

Statement of the inequality

Suppose n is a natural number and x 1 , x 2 , … , x n are positive numbers and:

n is even and less than or equal to 12, or

n is odd and less than or equal to 23.

Then the Shapiro inequality states that

∑ i = 1 n x i x i + 1 + x i + 2 ≥ n 2

where x n + 1 = x 1 , x n + 2 = x 2 .

For greater values of n the inequality does not hold and the strict lower bound is γ n 2 with γ ≈ 0.9891 … .

The initial proofs of the inequality in the pivotal cases n = 12 (Godunova and Levin, 1976) and n = 23 (Troesch, 1989) rely on numerical computations. In 2002, P.J. Bushell and J.B. McLeod published an analytical proof for n = 12.

The value of γ was determined in 1971 by Vladimir Drinfeld, who won a Fields Medal in 1990. Specifically, Drinfeld showed that the strict lower bound γ is given by 1 2 ψ ( 0 ) , where ψ is the function convex hull of f(x) = e^−x and g ( x ) = 2 e x + e x 2 . (That is, the region above the graph of ψ is the convex hull of the union of the regions above the graphs of f and g.)

Interior local mimima of the left-hand side are always ≥ n/2 (Nowosad, 1968).

Counter-examples for higher n

The first counter-example was found by Lighthill in 1956, for n = 20:

x 20 = ( 1 + 5 ϵ ,   6 ϵ ,   1 + 4 ϵ ,   5 ϵ ,   1 + 3 ϵ ,   4 ϵ ,   1 + 2 ϵ ,   3 ϵ ,   1 + ϵ ,   2 ϵ ,   1 + 2 ϵ ,   ϵ ,   1 + 3 ϵ ,   2 ϵ ,   1 + 4 ϵ ,   3 ϵ ,   1 + 5 ϵ ,   4 ϵ ,   1 + 6 ϵ ,   5 ϵ ) where ϵ is close to 0.

Then the left-hand side is equal to 10 − ϵ 2 + O ( ϵ 3 ) , thus lower than 10 when ϵ is small enough.

The following counter-example for n = 14 is by Troesch (1985):

x 14 = (0, 42, 2, 42, 4, 41, 5, 39, 4, 38, 2, 38, 0, 40) (Troesch, 1985)