Let be some norm on . The norm induces a metric, and the metric yields a notion of curve length: the supremum of sums of distances over partitions. The unit circle is a closed curve; how small can its length be under the norm?
For the Euclidean norm, the length of unit circle is . But it can be less than that: if is a regular hexagon, its length is exactly . Indeed, each of the sides of is a unit vector with respect to the norm defined by , being a parallel translate of a vector connecting the center to a vertex.
To show that cannot be beaten, suppose that is the unit circle for some norm. Fix a point . Draw the circle ; it will cross at some point . The points are vertices of a hexagon inscribed in . Since every side of the hexagon has length , the length of is at least .
It takes more effort to prove that the regular hexagon and its affine images, are the only unit circles of length ; a proof can be found in Geometry of Spheres in Normed Spaces by Juan Jorge Schäffer.