Let me clarify a definition first: A saddle point of a function occurs wherever two conditions are fulfilled:
1) the gradient vanishes
2) the Hesse matrix is indefinite
I conjecture the following theorem:
Let with be a twice differentiable function defined on a compact and convex -manifold . We have on the border and on the interior of . Then we have the following implication:
has at least two local maxima has at least one saddle point
Do you know if this is true? And how it can be proven?