Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

It is known that, generically, the convex hull of a hypersurface embedded in $\mathbb{R}^n$ has a panel structure of simplices. Of course one can construct embeddings where this is not the case, but these are "rare" in a sense that can be made precise. (For intuition, consider how difficult it is to build a chair with 4 legs that all touch the ground at the same time).

Details are in the following paper: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.97.4388&rep=rep1&type=pdf

Also, see the following image from the above paper: Panel structure of the convex hull of a hypersurface

Since the biconjugate (double application of the Legendre-Fenchel transform) is the closed convex hull of the epigraph of the original function, I was wondering if a suitable generalization could be made to the infinite dimensional case. Under certain conditions can a "panel structure" be put on epi F**?

Certainly one could not hope to build epi F** by sweeping out panels with finite dimensional simplices, but if a simplex is generalized to mean the convex hull of a collection of suitably separated vectors then perhaps something can be said.

Does anyone know if this has been studied, and if so what the topic is called and what the key papers are? Or, alternatively, is there something really obvious I'm missing - an immediate reason why such an idea could never work in the infinite dimensional case?

Edit: From private correspondence with an expert in the field, this is probably an open question (depending on precisely how the panel structure and "rarity" are defined). He suggested that a proof might rely on Smale's infinite dimensional generalization of Sards theorem.

share|improve this question

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.