A regular tetrahedron (in ) can be thought of as the convex hull of four pairwise non-adjacent vertices of a regular cube. A bisecting plane parallel to a face of the cube intersects the tetrahedron in a square (one can think of this as the product of two intervals, contained as the middle slice of the join of two intervals). A plane bisecting the long diagonal of a regular cube intersects the cube in a regular hexagon. In each case, the “slice” one obtains is “rounder” (in some sense) than the original pointy object.
The unit ball in the norm on is a “diamond”, the dual polyhedron to an -cube (which is the unit ball in the norm). In three dimensions, the unit cube is an octahedron, the dual of an (ordinary) cube. This is certainly a very pointy object — in fact, for very large , almost all the mass of such an object is arbitrarily close to the origin (in the ordinary Euclidean norm). Suppose one intersects such a diamond with a “random” -dimensional linear subspace . The intersection is a polyhedron, which is the unit ball in the restriction of the norm to the subspace . A somewhat surprising phenomenon is that when is very big compared to , and is chosen “randomly”, the intersection of with this diamond is very round — i.e. a “random” small dimensional slice of looks like (a scaled copy of) . In fact, one can replace by here for any (though of course, one must be a bit more precise what one means by “random”).
We can think of obtaining a “random” -dimensional subspace of -dimensional space by choosing linear maps and using them as the co-ordinates of a linear map . For a generic choice of the , the image has full rank, and defines an -dimensional subspace. So let be a probability measure on , and let define a random embedding of into . The co-ordinates of determine a finite subset of of cardinality ; the uniform probability measure with this subset as support is itself a measure , and we can easily compute that . For big compared to , the measure is almost surely very close (in the weak sense) to . If we choose to be -invariant, it follows that the pullback of the norm on to under a random is itself almost -invariant, and is therefore very nearly propotional to the norm. In particular, the pullback of the norm on is very nearly equal to (a multiple of) the norm on , so (after rescaling), is very close to an isometry, and the intersection of with the unit ball in in the norm is very nearly round.
Dvoretzky’s theorem says that any infinite dimensional Banach space contains finite dimensional subspaces that are arbitrarily close to in given finite dimension . In fact, any symmetric convex body in for large depending only on , admits an -dimensional slice which is within of being spherical. On the other hand, Pelczynski showed that any infinite dimensional subspace of contains a further subspace which is isomorphic to , and is complemented in ; in particular, does not contain an isometric copy of , or in fact of any infinite dimensional Banach space with a separable dual (I learned these facts from Assaf Naor).