A sphere is convex.
The word "sphere" is used in two different senses. Sometimes it means a solid ball, and sometimes it means the surface of a ball. In mathematics, it's better to use the word "ball" for the solid ball, and to reserve the word "sphere" for the surface of the ball. In that case, the sphere is not convex. Of course, the ball is convex.
A similar thing happens in two dimensions, where the word "circle" is sometimes used to mean a disk, and sometimes (more accurately) it refers to the perimeter of the disk.
I would like to see how one shows that a sphere in an arbitrary (not necessarily ) normed space is non-convex.
I would say that this is simply due to the fact that for the function is non-constant on . But is it really an adequate argument? There seems to be too much of intuition coming from in it, don't you find?