Space formed by dot products of three vectors Suppose I have 3 3D unit vectors $\mathbf{v}_1$, $\mathbf{v}_2$, and $\mathbf{V}$. I define the three corresponding scalars $u_1=\mathbf{v}_1 \cdot \mathbf{V}$,  $u_2=\mathbf{v}_2 \cdot \mathbf{V}$, and $U=\mathbf{v}_1 \cdot \mathbf{v}_2$, where obviously each take on values between -1 and 1. However, the range of each value is restricted by the other two, so the shape of the volume available to the system is quite strange. 
I'm interested in constructing functions of $u_1$, $u_2$, and $U$ that can only take on values between 0 and 1. To do that, it would be very useful to know more about this volume. Does anyone know if there is some existing work on this, or a useful search term? Failing that, is there a simple inequality that describes the allowed values of $u_1$, $u_2$, and $U$?
 A: Turned out not to be too hard to answer this after all. I found the following inequality:
$$u_1 u_2 - \sqrt{(1-u_1^2)(1-u_2^2)} \le U \le u_1 u_2 + \sqrt{(1-u_1^2)(1-u_2^2)}$$
This can be found by setting $\mathbf{V}=\{0,0,1\}$ and writing the other two vectors as 
$$\mathbf{v}_i = \{\sin{\theta_i}\cos{\theta_i}\,,\, \sin{\theta_i}\sin{\theta_i}\,,\, \cos{\theta_i}\}$$
A: Permit me the liberty of renaming your variables. We will consider three unit vectors $u$, $v$, and $w$, whose inner products are $u \cdot v = c$, $u \cdot w = b$, and $v \cdot w = a$.
Consider the Gram matrix of the vectors $u$, $v$, and $w$, namely the matrix of dot products
$$G =
\begin{bmatrix}
u \cdot u & u \cdot v & u \cdot w \\
v \cdot u & v \cdot v & v \cdot w \\
w \cdot u & w \cdot v & w \cdot w
\end{bmatrix}
=
\begin{bmatrix} 1 & c & b \\ c & 1 & a \\ b & a & 1 \end{bmatrix}.$$
A symmetric matrix $G$ is a Gram matrix if and only if it is positive semidefinite. Positive semidefiniteness is equivalent to the nonnegativity of all principal minors,
$$
1 - a^2 \ge 0, \\
1 - b^2 \ge 0, \\
1 - c^2 \ge 0, \\
1 - a^2 - b^2 - c^2 + 2 abc \ge 0.
$$
These are the conditions that define the volume you are interested in.
The shape looks like an inflated tetrahedron, all of whose axis-aligned slices are ellipses:

