TITLE: Will 3 lights illuminate any convex solid?
QUESTION [35 upvotes]: Can 3 lights be placed on the outside of any convex N dimensional solid so that all points on its surface are illuminated?

REPLY [3 votes]: Well it depends on how you decide to treat asymptotic behavior, i.e. - is the solid illuminated at the tangent of a ray? In a practical sense, because of light diffusion it would, but then your question seems pretty theoretical.
For $2$ dimensions:
$$x^2+y^2 = 1.$$
Let light source $1$ be at $(0, \infty)$, and source $2$ at $(0, -\infty)$.
This would basically make tangents with the circle at essentially $(1,0)$, and $(-1,0)$.
For $n$ dimensions:
$$x_1^2+x_2^2+ \cdots + x_n^2 =1.$$
It's basically the same, where you take a vector, place it at infinity: $(0,0, \dots, \infty)$, $(0,0, \dots, -\infty)$.
A similar behaviour with tangents occurs, although they'd happen along a region, rather than just a pair of points.
If you must shed light on those tangents, I guess things get a little more complicated, and $4$ sources of light would be your best bet.
This should be correct no matter which dimension, because light sources technically should radiate in all directions anyway, and it's really obvious given the fact that you can always break down the problem into 2d, and it'll be exactly the same, no matter where in the many dimensions you take it in.
Edit: This is mainly because if you place a point at $(0,0, \dots, \infty)$, if you look at it from a 2d perspective from any plane that does not include infinity, the light source basically shines directly onto the solid, since the point actually lies within the region.