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).