Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I'm looking for geometric applications of Ekeland's variational principle in order to see it at work in a context I'm familiar with. Let me recall the principle itself:

Definition. Let $(X,d)$ be a metric space, let $f$ be a real-valued function on $X$, and let $\epsilon > 0$. A point $m \in X$ is an $\epsilon$-minimizer if (1) $f(m) \leq \inf f + \epsilon$ and (2) $m$ is the unique minimizer of the perturbed function $x \mapsto f(x) + \epsilon d(m,x)$.

Ekeland's weak principle. If $f$ is lower semi-continuous and $(X,d)$ is complete, there is an $\epsilon$-minimizer for every value of $\epsilon > 0$.

Ekeland's strong principle. Assume $f$ is lower semi-continuous and $(X,d)$ is complete. If for a given $\epsilon > 0$ the point $y \in X$ satisfies $f(y) \leq \inf f + \epsilon$, then there is an $\epsilon$-minimizer $m \in X$ such that $f(m) \leq f(y)$ and which is at distance $\leq 1$ from the point $y$.

There are some applications in Ekeland's paper, but I'd like to see something more geometric using, for example, the Hausdorff distance on the space of convex sets. Actually, this principle caught my eye and I'm just curious as to what a geometer can do with it.

share|improve this question
add comment

Know someone who can answer? Share a link to this question via email, Google+, Twitter, or Facebook.

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.