For $\displaystyle X \subseteq E^3 $

and $\displaystyle u \in E^3$

we set

$\displaystyle S(u;X) = \left \{ tu + x: t > 0,x \in X \right \} $

$\displaystyle S(u;X)$ is the shadow of X when the light is coming from the direction u.

Let $\displaystyle u \in E^3$

.Prove .

(a) If X is a convex subset of $\displaystyle E^3$

, then $\displaystyle S(u;X)$ is also convex.

(b) If X is open in $\displaystyle E^3$

, then $\displaystyle S(u;X)$ is open and $\displaystyle X \subseteq S(u;X)$.

Im not sure how to aproach both of them.

Thanks.