On the area-perimeter ratio of a convex limited set (Previously asked on MSE)
Let $C\subset \mathbb{R}^2$ be a convex limited set. We define the average radius of $C$ as 
$$a_C=\frac{\int_{v\in C}d(v,C)dxdy}{A(C)}$$
Where $d(v,C)$ is the distance between the boundary of $C$ and $v$, and $A(C)$ is the area of $C$.
Are there constants $k_1> 0, k_2$ s.t. $k_1 \leq \frac{P(C)a_C}{A(C)}\leq k_2 $, where P(C) is the perimeter of $C$?
For example, if $C$ is a rectangle of sides $a\geq b$, the above ratio equals $\frac{(3ab^2-b^3)(a+b)}{6a^2b^2}$, which also proves that, if such constants exist, $k_1\leq 1/2$, $2/3\leq k_2$.
 A: Suppose that $C$ is a compact convex set on the plane. Let points $A,B$ in $C$ be such that $d(A,B)=\max\{d(u,v)\colon u,v\in C\}$. Then the line through the point $B$ perpendicular to the line $AB$ is a support line to $C$. Similarly, the line through the point $A$ perpendicular to the line $AB$ is a support line to $C$. Also, without loss of generality, the line $AB$ is horizontal, in some coordinate system. Let then $H$ and $L$ be, respectively, the highest and lowest points of $C$. Let (the quadrilateral) $Q$ be the convex hull of the set $\{A,L,B,H\}$. Let $R$ be the rectangle with the vertical sides through $A$ and through $B$, and with the horizontal sides through $L$ and through $H$. Then $Q\subseteq C\subseteq R$, whence 
$$\int_Q d(v,Q)\,dv\le\int_C d(v,C)\,dv\le\int_R d(v,R)\,dv\le c_1 \int_Q d(v,Q)\,dv
$$
for some universal real constant $c_1>0$. Also, 
$$Area(Q)\le Area(C)\le Area(R)=2\,Area(Q).
$$
Also,
$$P(Q)\le P(C)\le P(R)\le c_3 P(Q)
$$
for some universal real constant $c_3>0$. 
Now your desired result follows: the ratio $r(C):=\frac{P(C)a_C}{Area(C)}$ differs from $r(R)$ by, at most, universal positive real constant factors, and, as you showed,  $r(R)$ is bounded from above and from below by universal positive real constants. 

This solution is illustrated by the picture below, where the convex quadrilateral $Q$ is the darkest, the convex rectangle $R$ is the lightest, and the given convex set $C$ is of the intermediate darkness. 

