Shortest halving line

Question

Elsewhere I read a proof that a diameter is the shortest curve bisecting a circle (intuitively obvious), but an altitude is not the shortest bisector of an isosceles triangle (not obvious). Is anything known about the shortest bisector of an arbitrary shape (e.g. area/length^2 extrema, probably taken for a circle)? (Clearly the shape must be constricted to convex, otherwise the length can be zero.)