This question is related to my previous posting
Angle subtended by the shortest segment that bisects the area of a convex polygon
Let $C$ be a convex region in the plane and let $s$ be the shortest line segment that divides the area of $C$ in half, namely two pieces $C_1$ and $C_2$. Let $B$ be any ball whose center is contained in $C_1$. I am trying to prove that there exists a positive constant $\alpha$ such that, for any such $C$, $s$, and $B$, we have $$\frac{\mathrm{Area}(C_1\cap B)}{\mathrm{Area}(C\cap B)}\geq\alpha$$ Clearly it must be the case that $\alpha < 1/2$, and numerical simulations suggest that $\alpha>1/4$. Does this sound plausible? Are there any bisecting segments other than $s$ that might work instead?
