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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?

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  • $\begingroup$ Support function of a convex curve is a convex function of one variable, so it is absolutely continuous. Unless you have some other definition of support function. $\endgroup$ Feb 11, 2016 at 15:44

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