what is the meaning of asphericity of convex set in a linear normed space?

Question

I am trying to understand the definition of spherical to within $\epsilon$ for a convex body in a linear normed space, as given in Aryeh Dvoretzky's paper [1] (section 2, page 203): A convex set $C$ in a Euclidean space is called spherical to within > $\epsilon$ ($0 < \epsilon < 1$) if there exist in the flat space generated by $C$ two concentric balls $B_1$ and $B_2$ of radii $(1 — \epsilon)r$ and $r$, respectively, such that $B_1 \subset C \subset B2$. The greatest Iower bound of the $\epsilon$ having the above property is called the asphericity of $C$ and denoted by $a(C)$. I don't understand this definition very well since i don't know what he means by the flat space generated by $C$? and i need example ,if possible, clearing the definition.

[1] Aryeh Dvoretzky. Some near-sphericity results. Proc. Sympos. Pure Math. 7, 203-210 (1963).
AMS ebook: http://www.ams.org/books/pspum/007/0158308/pspum0158308.pdf
Google books preview: http://books.google.com/books?id=MuEFJR7Ek4EC&pg=PA203

Answer 1

Flat space: refer to "Flat_(geometry)" and "Euclidean_space".

I think he means, because he's talking about $n$-dimensional Euclidean space, that the set $C$ can exist in fewer than $n$ dimensions. If $C$ is $m$-dimensional, $m \leq n$, then the flat space generated by $C$ is the $m$-dimensional space containing $C$.

E.g. In $3D$ space, $C$ might be a circle (obviously with $2$ dimensions), which would be exactly "spherical" in this context, or an ellipse, which would be "spherical to within $\epsilon$", with the value of $\epsilon$ depending on how round the ellipse was.

In this case, $B_1$ is a ($2$-dimensional) disc fully inside $C$ and $B_2$ is a disc fully containing $C$, with $B_1$ and $B_2$ concentric. The "flat space generated by $C$" here is the plane containing $C$.

The asphericity, $a\left(C\right)$, is a measure of how far apart $B_1$ and $B_2$ must be for $C$ to lie between them.