Exercise 41, § 7

Let $S$ be a compact simplex contained in a Hausdorff locally convex space $E$ (II, p. 71 exerc. 41), and $f$ an upper semi-continuous convex function that is bounded below. Let $g$ be a lower semi-continuous function that is concave and such that $g \geq f$. a) If we write $u = \tilde{f}, v = -(-g)\tilde{f}$, then $u$ and $v$ are affine functions such that $u \leq v$ (use exerc. 39, b) and exerc. 40). b) Show that there exists an affine function $h$, continuous in $X$ and such that $f \leq h \leq g$ (D. Edwards’ th.). (We can replace $f$ by $u$ and $g$ by $v$. Construct three sequences $(u_m), (v_m), (h_m)$ of affine functions such that in $X$, the function $u_m$ is upper semi-continuous, the function $v_m$ is lower semi-continuous and the function $h_m$ is continuous and

Use exerc. 29 of II, p. 81, for this.)

EXERCISES