Geometric dilation of closed curves in normed planes

It is known that the lower bound for the geometric dilation of rectifiable simple closed curves in the Euclidean plane is π/2, which can be attained only by circles. We extend this result to (normed or) Minkowski planes by proving that the lower bound for the geometric dilation of rectifiable simple closed curves in a Minkowski plane X is analogously a quarter of the circumference of the unit circle SX of X, but can also be attained by curves that are not Minkowskian circles. In addition we show that the lower bound is attained only by Minkowskian circles if the respective norm is strictly convex.