How the following body in a Euclidean space $\mathbb{R}^n$: $$R(d;\phi) = \{v\in\mathbb{R}^n \mid \widehat{vd}<\phi, v\ne 0\}$$ is called (for some fixed nonzero vector $d$ and angle $\phi$, we can assume $\phi<\pi$)?
$\widehat{ab}$ here denotes the angle between vectors $a$ and $b$.
I feel it is related with cones and conical surfaces, but not sure about the precise term.
It is easy to see that $$R(d;\phi) = \{v \in \mathbb R^n \mid (v,d) > \cos(\phi) \, \|v\|\,\|d\|\}$$
In the special case $d = e_1$ and $\cos(\phi) = 2^{-1/2}$ you have $$R(d;\phi) = \{v \in \mathbb R^n \mid 2^{-1/2} \, \|v\| < v_1 \}\\ = \{v \in \mathbb R^n \mid \|\hat v\| < v_1\},$$ where $\hat v = (v_2, \ldots, v_n) \in \mathbb{R}^{n-1}$. In convex optimization, the closure of this special $R(d;\phi)$ is called second-order cone.
Hence, one could call your set open, generalized second-order cone.