# Convex-like properties of the polar parametrization of the boundary a convex body on the plane

Let $$B$$ be a compact convex set on the complex plane, containing zero in its interior. The boundary $$\partial B$$ of $$B$$ has the polar parametrization $$\mathbf p:\mathbb R\to \partial B$$ assigning to each real number $$t$$ the unique point of the intersection of $$\partial B$$ with the ray $$\{re^{it}:r>0\}$$. The function $$\mathbf p$$ is $$2\pi$$-periodic and has two main properties of convex functions: it has two-sided derivatives $$\mathbf p'_-(t)$$ and $$\mathbf p'_+(t)$$ for every $$t\in \mathbb R$$ and those two-sided derivatives have bounded variation on any bounded subset of the real line. Consequently $$\mathbf p$$ is twice differentiable almost everywhere. Those facts are more-or-less "obvious" but I do not see short proofs (or even short reduction to the known properties of convex functions). Maybe this is alredy done somewhere? Just to insert a reference and not spend time for reproving known results. Please, help.

If $$f(x)=\min\{s>0:x/s\in B\}$$ is Minkowski functional of $$B$$, then $$f$$ is a convex function on the plane and $${\bf p}(t)=\frac{e^{it}}{f(e^{it})}$$. I think your claims now follow from the properties of convex functions. For example, you may locally choose a smooth function $$h(t)\in (0,\infty)$$ such that $$h(t)e^{it}$$ runs over a segment $$[AB]$$ on the plane when $$t$$ runs over a small enough interval $$\Delta$$, then write $${\bf p}(t)=\frac{h(t)e^{it}}{f(h(t)e^{it})}$$, and choose the parameter on $$[AB]\ni h(t)e^{it}$$ as a new variable.
• Thank you very much for the suggestion. I tried to use the convexity of the ball in an affine system and had problems with the proof of the boundedness of variation of the (one-sided) derivatives of $\mathbb p$. With your "Minkowski" approach this problem does not arize. So, thank you once more. – Taras Banakh Oct 30 '19 at 9:27