MathDB

Let

F:\mathbb{R}^2\to\mathbb{R}

and

g:\mathbb{R}\to\mathbb{R}

be twice continuously differentiable functions with the following properties:
•

F(u,u)=0

for every

u\in\mathbb{R};

• for every

x\in\mathbb{R},g(x)>0

and

x^2g(x)\le 1;

• for every

(u,v)\in\mathbb{R}^2,

the vector

\nabla F(u,v)

is either

\mathbf{0}

or parallel to the vector

\langle g(u),-g(v)\rangle.

Prove that there exists a constant

C

such that for every

n\ge 2

and any

x_1,\dots,x_{n+1}\in\mathbb{R},

we have

\min_{i\ne j}|F(x_i,x_j)|\le\frac{C}{n}.

Problem Statement