There is an example of a convex function of two variables that is convex but not absolutely continuous. The level set of this function is a convex curve.
To construct one example of such a function one needs to take a Cantor ladder $c$ and define $f(x,y) = g(x+y)$ where $g(x) = \int_0^x c(t) dt$.
Is it possible to give a example of a convex curve whose support function is not absolutely continuous?
