2
$\begingroup$

A $k$-dimensional section of a convex body $K \subset {\mathbb R}^n$ is just the intersection of $K$ with a $k$-dimensional hyperplane $h$.

Such a section is said to be $(1+\epsilon )$-almost spherical if $B(0,\frac {R}{1 + \epsilon }) \subset h \cap K \subset B(0, (1 + \epsilon )R)$, where $B(0,R)$ denote the Euclidean ball of radius $R$ about the origin.

Dvoretzky's theorem states that any centrally symmetric convex body $K \subset {\mathbb R}^n$, with non-empty interior contains a $k$-dimensional section which is $(1 + \epsilon )$-almost spherical, provided $n \geq n_0(k,\epsilon )$.

My question is about the function $n_0(k,\epsilon )$. Milman proved that $n_0(k,\epsilon ) \leq \epsilon ^{-ck\epsilon ^2}$, for some constant $c>0$. Gordon later improved this to $n_0(k,\epsilon )\leq 2^{ck/\epsilon ^2}$. Is this dependence on $\epsilon $ tight?

$\endgroup$

1 Answer 1

2
$\begingroup$

Schechtmann replaced the $\varepsilon^2$ in Gordon's bound by $\varepsilon$

Two observations regarding embedding subsets of Euclidean spaces in normed spaces, Advances in Mathematics 200(1), 125-135 (2006), doi:10.1016/j.aim.2004.11.003

Euclidean Sections of Convex Bodies, In Asymptotic Geometric Analysis, Volume 68 of the series Fields Institute Communications, 271-288 (2013), doi:10.1007/978-1-4614-6406-8_12, arxiv:1110.6401

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.