Return to Answer

4. On the upper half-space take the usual Euclidean norm, on the lower half-space take the norm whose unit ball is a cone with base equal to the unit disc on the wall and apex $(0,0,\dots,0,-1)$. You can verify that all light rays coming from the upper-half plane get refracted into the sheaf of vertical rays.

4. On the upper half-space take the usual Euclidean norm, on the lower half-space take the norm whose unit ball is a double cone with base equal to the unit disc on the wall and apexes $(0,0,\dots,0,1)$ and $(0,0,\dots,0,-1)$. You can verify that all light rays coming from the upper-half plane get refracted into the sheaf of vertical rays.

2. When condition (b) cannot be met is exactly the case where we have a "critical angle" and light is reflected instead of refracted. A good exercise is to check these constructions in the standard case as rediscover the usual laws of reflection and refraction that you find in any physics textbook.

2. When condition (b) cannot be met is exactly the case where we have a "critical angle" and light is reflected instead of refracted. A good exercise is to check these constructions in the standard case and rediscover the usual laws of reflection and refraction (and the condition for critical angles) that you find in any physics textbook.

This is a good time to say that what I'm writing now is basically an annoucement of an essay I'm writing for a book with A.C. Thompson ("An invitation to Minkowski geometry"). What I'm writing now is a bit sketchy at some points (otherwise it would be too long).

(c') The points at which the hyperplane $\eta = 1$ supports the unit ball $B_2 = \{ v : \|v\|_2 \leq 1 \}$ lie and the points at which the hyperplane $\xi = 1$ supports the unit ball $B_1 = \{ v : \|v\|_2 \leq 1 \}$ lie on different sides of $W$.

This is a good time to say that what I'm writing now is basically an annoucement of an essay I'm writing for a book with A.C. Thompson ("An invitation to Minkowski geometry"). The presentation here is a bit sketchy at some points (otherwise it would be too long), but if you reconstruct the pictures, I think everything will be clear.

(c') The points at which the hyperplane $\eta = 1$ supports the unit ball $B_2 = \{ v : \|v\|_2 \leq 1 \}$ and the points at which the hyperplane $\xi = 1$ supports the unit ball $B_1 = \{ v : \|v\|_2 \leq 1 \}$ lie on different sides of $W$.