Figure of merit

In mathematics, the Shapiro inequality is an inequality proposed by H. Shapiro in 1954.

Statement of the inequality

Suppose n is a natural number and $x_1,x_2,\dots ,x_n$ 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

$\sum _{i=1}^n\frac{x_i}{x_{i+1}+x_{i+2}}\ge \frac{n}{2}$

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 $\gamma \frac{n}{2}$ with $\gamma \approx 0.9891\dots$.

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 $\frac{1}{2}\psi (0)$, where ψ is the function convex hull of f(x) = e^−x and $g(x)=\frac{2}{e^x+e^{\frac{x}{2}}}$. (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:

$x_{20}=(1+5ϵ,6ϵ,1+4ϵ,5ϵ,1+3ϵ,4ϵ,1+2ϵ,3ϵ,1+ϵ,2ϵ,1+2ϵ,ϵ,1+3ϵ,2ϵ,1+4ϵ,3ϵ,1+5ϵ,4ϵ,1+6ϵ,5ϵ)$ 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)

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External links

• Usenet discussion in 1999 (Dave Rusin's notes)
• PlanetMath