Let $D$ be a convex region, and let $x \in D$. for every unit vector $y$, let $d_y(x, \partial D)$ denote the distance from x to $\partial D$, measured along the ray connecting $x$ to $y$. Then it is a proposition that $\int_{\left \| y \right \|=1} \frac{dy}{d_y(x, \partial D)}$ is constant in $x$.
Since we sum over all possible directions and meet every point on $\partial D$ exactly once, this seems equivalent to the integral $\int \limits_{\partial D} \frac{dy}{d(x, y)}$. This integral looks "physical":
If we consider $D$ to be a surface of equal electric charge, and $x$ is an electron inside, Then the potential in $x$ due to a point $y$ on the boundary is $\frac{1}{d(x,y)}$ up to multiplication by a constant. Thus I want to interpret this integral as the total potential in $x$, and the claim becomes: The potential inside any convex body of uniform charge is constant. By the divergence theorem (Gauss' theorem), the electric field inside $D$ is zero, and hence the potential, being its integral, is constant.
There are numerous hand-wavy claims in the above: is it truly a proof?
EDIT User @Rahul pointed that the electric field being zero in the interior is not true for general $D$, only for spheres. So in the case of a sphere does this reasoning work? And if so, can it be salvaged in a more general setting?