# Helly-type theorem for infinite-dimensional spaces

Currently I'm reading a classical text Convex Sets by F. Valentine. While reasing it I'm trying to generalize results as much as I can. I'm a bit confused of authors use of a notions of Euclidean space. Although it is defined at page 57 as finite dimensional real inner product space (a standard definition), Valentine keeps using a notion of closed affine subspaces instead of just affine subspaces and closed and bounded sets instead of compacta. This suggests that theorems are intended to be proved for some classes of topological vector spaces space $$\mathsf{X}$$: I think, I managed to render some result related to convex cones and proved by Valentine for 'Euclidean Spaces' for locally convex reflexive spaces.

If $$\mathsf{X}$$ actually contain infinite-dimensional spaces, one of such theorems which can be considered an infinite-dimensional generalization of Helly theorem:

Let $$(C_i)_{i \in I}$$ be a collection of closed and bounded convex sets in a space $$V\in\mathsf{X}$$, then the following is equivalent, let $$n \in \mathbb{N}$$: for each finite collection $$\{i_1,\ldots,i_n\}\subset I$$ the intersection $$\bigcap^n_{j=1} C_{i_j}$$ is non-empty $$\iff$$ every closed affine subspace $$A$$ with $$\mathrm{codim} \; A = n - 1$$ has a translate which intersects every $$C_i$$.

I think this is a Helly-type theorem, because it can be restated as, a family of sets $$C_i$$ as described above is an $$n$$-Helly family $$\iff$$ $$\left[\bigcap_{i \in I} \pi_{n-1}^* C_i \right]= \mathbf{Gr}^*(V,n-1)$$, where $$\pi_k^* : 2^V \to 2^{\mathbf{AGr}^*(V,k)}$$ is a mapping to subsets of 'affine Grassmanian' of closed affine subspaces of codimension $$k$$. In case $$\dim V = n-1$$ the dimension of these spaces is $$0$$, so they are just points, hence, we get a normal Helly's theorem!

I think this result is also correct for semi-reflexive locally convex spaces.

My question: is it possible that the theorem holds for any infinite dimensional spaces? If this is the case, what is the best description of the class $$\mathsf{X}$$?

• Did you check Hörmander's book "Notions of Convexity", published by Birkhauser? – Bazin Feb 15 at 22:37
• @Bazin Thank you for your suggestion. I never thought that this book [Hörmander] tackles Helly-type theorems or results proved with convex cones and I thought it was something connected with advanced convex and harmonic analysis. Nevertheless, I will check it for the answers. – Nik Pronko Feb 15 at 22:56
• It seems that you forgot the convexity assumption. And there is some mess with the indices. – Jochen Wengenroth Feb 16 at 6:19
• Yes, the convexity is important – Nik Pronko Feb 16 at 6:54