I am having trouble proving the following:(adsbygoogle = window.adsbygoogle || []).push({});

Suppose that [itex]E[/itex] is a convex region in the plane bounded by a curve [itex]C[/itex]. Show that [itex]C[/itex] has a tangent line except at a countable number of points.

[itex]E[/itex] is convex iff for every [itex]x, y \in E,[/itex] and for every [itex]\lambda \in [0,1], (1-\lambda) x + \lambda y \in E[/itex].

I am considering an approach where I parametrize [itex]C[/itex] in a fixed orientation and then look at the places where it is not differentiable, showing somehow that corners with some angular measure [itex]a \in [0,\pi)[/itex] are the only flavor of non-differentiable parts on this curve, and then showing that the number of corners is bounded by [itex]\frac{2\pi}{\pi - a}[/itex] for the largest [itex]a[/itex].

Any thoughts?

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# Tangents of a Convex Region

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