Let $P$ be almost a convex polygon in the plane. By almost, we mean that some edges could be arcs and not just line segments, as long as $P$ remains convex. Let $L$ be a line going through $P$, dividing $P$ into two regions, $P_1$ and $P_2$.
Show that both $P_1$ and $P_2$ are convex.
This problem comes from the idea of dividing a circle by chords, in which case the resulting regions are polygons or polygons with one arc instead of line segment. But I believe in this more general setting, the same proof would hold.
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$\begingroup$More generally: the intersection of two convex sets is convex. This is immediate from the definition.
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