Q:

The ratio of the volume of a trihedron in a ball to the volume of that ball

In $\mathbb{R}^3$ consider the convex body defined as 
  $$C=\left\{(x, y,z): x \geq 0, y \geq 0, z-x-y \geq 0\right\}.$$

Let $B(0,r)$ denote the open ball of radius $r>0$ with the center at zero. I am thinking about the following questions: 
a)  Is it true that the ratio $\frac{V(C\cap B(0,r))}{V(B(0,r))}$ does not depend on $r$?
(My conjecture is that this is not necessarily so.)
b)  If a) is not true then is it possible to show that the ratio $\frac{V(C\cap B(0,r))}{V(B(0,r))}$ is bounded from below by a positive constant for any $r>0$?
(My conjecture would be yes but I don’t know how to show this.)
c)  Can the conclusion either in a) or b) be reached for any convex body $C$ with non-empty interior in $\mathbb{R}^3$ obtained by the intersection of a finite number of half-spaces containing $(0,0,0)$? 
I would greatly appreciate any hints on how to possibly approach these questions.  

A:

As achille hui explained, the scaling transformation $x\mapsto \frac{x}{r}$ can be used to answer this, and in greater generality. 
Let $C$ be any cone, meaning a set such that $x\in C\implies rx\in C$ for all $r>0$. All sets $C$ you consider are cones in this sense. 
The scaling transform $x\mapsto x/r$ maps $C\cap B(0,r)$ onto $C\cap B(0,1)$. The Jacobian determinant of this transform is $r^n$. Hence, 
$$\frac{\operatorname{vol}(C\cap B(0,r)}{\operatorname{vol}(C\cap B(0,1)} = r^n$$

