Suppose we have a 3-D shape $S$ with a center $C$, so that a point $p$ is in $S$ if and only if for any direction $\vec d$, $p$ is contained within a cylinder of radius $1$, extending infinitely both ways in the direction $\vec d$ and with its axis passing through $C$.
Is the resulting shape a sphere of radius $1$?
Assuming that you’ve seen that the sphere of radius $1$ is contained in every such cylinder, I’ll give you this hint: consider a point $P$ that's not in the unit sphere. Can’t you find one of your cylinders that similarly doesn’t contain $P$?
To focus your mind, you might coordinatize lightly, putting your center $C$ at the origin of Cartesian $3$-space. And permit yourself to choose your bad point with very good coordinates.