Conjecture about expected distance between two points in a regular polygon

Question

Point $P$ is a uniformly random point on the perimeter of a regular polygon.
Point $C$ is a fixed point somewhere on the circle inscribed in the polygon.
Is the following conjecture true:

The expected distance between $C$ and $P$ is maximized when $C$ is where the circle touches the polygon.

Numerical investigation shows that the conjecture is true for a square.

Suppose the square has vertices $(\pm1,\pm1)$. $C$ has coordinates $(\cos\theta, \sin\theta)$.

$E(PC)=\frac18\left(\int_{-1}^1 \sqrt{(\cos\theta-x)^2+(\sin\theta+1)^2}dx+\int_{-1}^1 \sqrt{(\cos\theta-1)^2+(\sin\theta-y)^2}dy+\int_{-1}^1 \sqrt{(\cos\theta-x)^2+(\sin\theta-1)^2}dx+\int_{-1}^1 \sqrt{(\cos\theta+1)^2+(\sin\theta-y)^2}dy\right)$

Here is the graph of $E(PC)$ against $\theta$ for $0\le \theta \le \pi/2$.

The graph shows that $E(PC)$ is maximized when $\theta=0$, i.e. when $C$ is where the circle touches the square.

Is my conjecture true for all regular polygons?

Answer 1

I wrote a script to estimate $E(PC)$ for various choices of $C$.

Let $n$ be the number of sides. It appears that for $n \leq 5$, your conjecture is true and that for $n\geq6$, $E(PC)$ is maximised when $C$ is collinear with the center and a vertex. I have no idea why.

Here's the Python code if it's useful to anyone:

from numpy import cos,pi,exp,mean,argmax
M=2000

for n in [3,4,5,7,8]:
    d=1/cos(pi/n)
    verts = [d*exp(2j*pi*k/n) for k in range(n)]+[d]
    ps=[i/M*verts[k]+(1-i/M)*verts[k+1] for i in range(M) for k in range(n)]
    cs=[exp(a/10*pi*1j/n) for a in range(11)]

    print(n,argmax([mean([abs(c-p) for p in ps]) for c in cs]))