# Geometric applications of Ekeland's variational principle

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.

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## 1 Answer

there are some examples in convex analysis. For example, in the book of Borwein & Lewis (Convex Analysis and nonlinear optimization) page 225 they use the PVE to prove that every Chebyshev set (i.e. every set that has the property "every point has a unique nearest point") is in fact convex (in finite dimensional spaces). The problem in arbitrary Banach spaces is open (as far I know). In general, I think that if you can to study some property of a set and you are able to associate a lsc function bounded from below function then you can use the PVE to get some properties of the set. This is specially useful because this framework is more adequate to work with nonsmooth function. In fact, you can take the distance function to a set and define some class of set saying some property of this function. This has been done to define, for example, the class of regular sets, prox-regular sets, etc.

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