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.
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