A sphere 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?