Cutting a sandwich with a crust Let $S$ be a simple closed curve in ${\Bbb R}^2$ enclosing a convex region $I$. Must there exist a straight line which cuts $S$ into two pieces of equal length and also cuts $I$ into two regions of equal area? If so, how can such a line be found? [If the answer is, "no, because the sandwich might have a pathological boundary", then please also consider the case of non-pathological sandwiches.]

Let $$s\mapsto z(s)\qquad(0\leq s\leq L)$$ be the counterclockwise representation of the crust with respect to arc length. Hold a knive over the sandwich connecting the points $z(0)$ and $z(L/2)$, and assume that the area to the right of the knive is more than half of the sandwich. Now turn the knive slowly counterclockwise so that at all times it points from $z(s)$ to $z(s+L/2)$. When we arrive at $s=L/2$ we will have less than half of the sandwich to the right of the knive. By the intermediate value theorem there has to be a position $\sigma\in\ ]0,L/2[\ $ of the knive for which the area of the sandwich is exactly halved.