Is the support function continuous on its effective domain?

Question

Let $\sigma_D(x)=\sup \{ \left< x, y \right> : y\in D \}$ for a closed convex $D\subseteq \mathbb R^n$. Then $\sigma_D$ is convex and lower semicontinuous (it's the supremum of linear functions). Let the effective domain of $f$ be the set of points where $f$ is finite. Then $\sigma_D$ restricted to its effective domain is continuous at interior points as well as at polyhedral boundary points by the Gale-Klee-Rockefallar theorem.

I came across an unsourced claim in an economics paper that $\sigma_D$ restricted to its effective domain is continuous, but have not been able to construct or source a proof. Is that claim correct? If not, what's a counterexample?

Answer 1

The answer is negative as can be seen by putting together these two facts:

1. There is a bounded convex lower semicontinuous functions defined on a closed and convex subset of $\mathbb R^2$ that is not continuous. One example is here. For a large stock of examples, see this.

2. Every convex, lower semicontinuous and positively homogeneous function on $\mathbb R^n$ is a support function. Theorem 13.2 in Rockefellar's Convex Analysis.

For then by (1), let $f$ be a non-continuous bounded convex lower semicontinuous function on a closed subset $D$ of the plane. Extend $f$ to all of the plane by setting it to $+\infty$ outside $D$. This is still convex and lower semicontinuous and it's not continuous on its effective domain.

Finally, define $g$ on $\mathbb R^3$ as follows: $$ g(x,y,z) = \begin{cases} zf(x/z,y/z) &\text{if }z>0\\ +\infty & \text{if }z\le 0\text{ and }(x,y)\ne 0\\ 0 &\text{if }(x,y,z)=0. \end{cases} $$ This is convex and lower semicontinuous everywhere and not continuous on the effective domain, and it's positive homogeneous. By (2), $g$ is a support function.