  In this note we investigate some aspects of the local structure of finite
dimensional $p$-Banach spaces. The well known theorem of Gluskin gives a sharp
lower bound of the diameter of the Minkowski compactum. In [Gl] it is proved
that diam$({\cal M}_n^1)\geq cn$ for some absolute constant $c$. Our purpose is
to study this problem in the $p$-convex setting. In [Pe], T. Peck gave an upper
bound of the diameter of ${\cal M}_n^p$, the class of all $n$-dimensional
$p$-normed spaces, namely, diam$({\cal M}_n^p)\leq n^{2/p-1}$. We will show
that such bound is optimum.
