Geometric description of set

Question

What would be a geometric description of the following set: $$D=\{u \in \mathbb{R}^n ,r\leq |u-x/2| \leq R , |(u,x)|\leq C \}$$ Where $x\in \mathbb{R}^n $ is a fixed point, $C>0$. It looks like part of a spherical wedge if i am not mistaken . any thoughts ?

Answer 1

Let $x\in\mathbb{R}^n$, and let $r,R,C\in\mathbb{R}^+$. The set $$ A=\{u\in\mathbb{R}^n : r\leq|u-x/2|\leq R\} $$ is an annulus centered at $x/2$, and the set $$ S=\{u\in\mathbb{R}^n : |(u,x)|\leq C\} $$ is a "strip" of "width" $\frac{2C}{|x|^2}$ centered at the origin running in all directions orthogonal to $x$. The set $D$ in the OP is the intersection of these two sets. Depending on the choices of all parameters involved you will get many different kinds of shapes, so I think this is the most that can be said.