2 Changed "at least" to "at most".

Here is a problem in convex geometry. Qualitatively it asks: If a finite dimensional normed space has the property that an arbitrary subspace is approximately Euclidean after throwing away a small number of dimensions, must the space itself be approximately Euclidean?

Here is the question:

Fix a constant $C>1$; let's use $C=10$. Suppose that $B$ is a convex symmetric body in $\mathbb{R}^n$ such that for every subspace $F_1$ of $\mathbb{R}^n$ there is a subspace $F$ of $F_1$ of codimension at least most $\log \log \dim F_1$ in $F_1$ and a centrally symmetric ellipsoid $E$ in $F$ with $E\subset B\cap F \subset 10 E$. The question is whether there is a constant $\gamma$, independent of everything except the constant $C=10$, so that there is an ellipsoid $E \subset \mathbb{R}^n$ with $E\subset B \subset \gamma E$.

Can you answer this question using only finite dimensional considerations? AFAIK, the only answers use infinite dimensional tools.

1

Here is a problem in convex geometry. Qualitatively it asks: If a finite dimensional normed space has the property that an arbitrary subspace is approximately Euclidean after throwing away a small number of dimensions, must the space itself be approximately Euclidean?

Here is the question:

Fix a constant $C>1$; let's use $C=10$. Suppose that $B$ is a convex symmetric body in $\mathbb{R}^n$ such that for every subspace $F_1$ of $\mathbb{R}^n$ there is a subspace $F$ of $F_1$ of codimension at least $\log \log \dim F_1$ in $F_1$ and a centrally symmetric ellipsoid $E$ in $F$ with $E\subset B\cap F \subset 10 E$. The question is whether there is a constant $\gamma$, independent of everything except the constant $C=10$, so that there is an ellipsoid $E \subset \mathbb{R}^n$ with $E\subset B \subset \gamma E$.

Can you answer this question using only finite dimensional considerations? AFAIK, the only answers use infinite dimensional tools.