The optimization problem I am trying to solve is given by
$$\min_{f \in C^2[a,b]} \,\, \inf_{x \in [a,b]} f(x)$$ subject to the constraints $$\int_a^b\sqrt{1+f'(x)^2}\,dx = L$$ , $f(a) = A$ and $f(b) = B$. Here $L,a,b,A,B$ are some constants with $a < b$ and $L$ is greater than the length of the line segment between $(a,A)$ and $(b,B)$.
This is a variation of the well-known catenary problem in which the objective is $$\min_{f \in C^2[a,b]}\int_a^bf(x)\sqrt{1+f'(x)^2}\,dx$$ and the constraints are the same as the ones above.
From a physical point of view what I am wondering is can I distort a chain (one that cannot be elongated) whose end points are fixed such that the lowest point of the chain is lower than what it would be if I let gravity do its thing and did not meddle with the chain at all. My intuition is that it is not possible. If I take a chain hanging in equilibrium from two points on the plane and pull it to the side, its lowest point would move up I think.
I can write $$\int_a^bf(x)\sqrt{1+f'(x)^2}\,dx \geq \inf_{x\in[a,b]}f(x)\int_a^b\sqrt{1+f'(x)^2}\,dx = \inf_{x\in[a,b]}f(x)L$$
So the objective of the catenary problem provides an upper bound for the objective of my problem but I don't see if that helps.
Hint:
The minimum is attained if the chain is completely strained. At this case the locus of the infimum is an ellipse whose foci are on $[a,f(a)]$ and $([b,f(b)])$.