Show that $N$ is an Euclidean norm if and only if the intersection of the unit ball with any plane is an ellipse.
I'm stuck on this one. I do not see how can I connect the definition of an Euclidean norm and the geometric appearance requested. I know that $N$ is Euclidean if and only if $\|v+u\|=\sqrt{2\|v\|^2+2\|u\|^2-\|v-u\|^2}$
Any help is appreciated.
The formula you wrote combines the two halves of the answer. It shows that a norm is Euclidean iff it's restriction to each 2-d linear subspace is a Euclidean norm. Now, see if you can prove that a norm on a 2-plane is Euclidean iff the unit ball is an ellipse. (Use the same formula.)