6 1. THE RESULT

It has been an open problem in the last 25 or so years if this is true for non-simple

polytopes (even for a single one!), and it is precisely what Theorem 1.1 claims in a

slightly sharper form. The second term on the right of (1.16) is usually dominated

by the first one, so the main improvement in Theorem 1.1 is not the dropping of

this term (although we shall see that dropping that term is an important step in

the proof), but the dropping of the “simple polytope” assumption. Why are simple

polytopes easier to handle, i.e. why is (1.14) for simple polytopes substantially

weaker then for general ones? The answer is that the crux of the matter is approxi-

mation around the vertices of the polytope. Now a vertex of a simple polytope looks

like a vertex of a cube (modulo an aﬃne transformation), and cubes are relatively

easy to handle since they are products of segments (therefore, approximation on

cubes can be reduced to approximation on [−1, 1], as was done in [12]). This is no

longer true if there are more than d edges at a vertex. Still, the simple polytope

case will play an important role in the proof of Theorem 1.1.