Proposition 21.11.45C0908AA2E3B2

Proof. Apply the proved decreasing smooth approximation with fixed growth body. It gives smooth convex $F_j$ with bounded difference from the support function of $P$, decreasing pointwise, in fact locally uniformly, to $F$. The bounded-difference condition is precisely membership in $\mathcal E^\infty(N,P)$, and $\mathcal P(N,P)$ is the same upper-growth class in both formulations. That construction preserves the common growth body $P$, including when $P$ has empty interior; therefore the empty-interior case requires no full-dimensional approximation of $P$.