Distortion from spheres into Euclidean spaces

Any function from a round $n$-dimensional sphere of radius $r$ into $n$-dimensional Euclidean space must distort the metric additively by at least ${\displaystyle \frac{\pi r}{1+\sqrt{1-\frac{2}{n+2}}}}$ if $n$ is even and ${\displaystyle \frac{\pi r}{1+\sqrt{1-\frac{2(n+2)}{(n+1)(n+3)}}}}$ if $n$ is odd. This is proved using a fixed-point theorem of Granas that generalizes the classical theorem of Borsuk-Ulam to set-valued functions.