Exercise 2, § 6

Let $D$ be a closed convex set in a Banach space $F$, with nonempty interior, and let $f$ be a measurable function with values in $F$, such that $f(X) \subset D$. Let $g$ be an integrable function $\geq 0$, not negligible, and such that $fg$ is integrable. Suppose that the point

is a boundary point of $D$; show that if $V$ is the intersection of all the closed support hyperplanes of $D$ at the point $c$, then $f(x) \in V \cap D$ almost everywhere in the set of points $x$ such that $g(x) > 0$ (reduce to the case that $F$ is a separable space (that is, contains a countable dense set), using Th. 4 of §5; then note that $V$ is the intersection of a countable family of support hyperplanes of $D$ at the point $c$).

Show that if $F$ is finite-dimensional and if $A$ is the facet of the point $c$ with respect to $D$ (TVS, II, §7, Exer. 3), the hypothesis implies that $f(x) \in A$ almost everywhere in the set of points $x$ such that $g(x) > 0$ (argue by induction on the dimension of $F$).