The title may seem a bit confusing, let's use math notation.
Let $c:\mathbb{R}\to \mathbb{R}^2$ be a simple closed curve parametrized by length. A pair of points on the curve that bisect the perimeter means any pair in the form $(c(t),c(t+\frac{L}{2}))$ where $L$ is the total length of $c$. Now I am interested in the smallest distance between those pairs, i.e. $$\inf_{t\in[0,{L/2})}|c(t)-c(t+L/2)|.$$ Let's call this smallest bisection length. Now it seems that fixing $L$, the shape that maximizes the smallest bisection length is the circle with radius $L/(2\pi)$. So, is this true?
Any help will be appreciated! Another question may be related to this is what shape will give the maximum area given a fixed perimeter, whose answer is a circle.