Circles minimize most knot energies

We define a new class of knot energies (known as renormalization energies) and prove that a broad class of these energies are uniquely minimized by the round circle. Most of OʹHaraʹs knot energies belong to this class. This proves two conjectures of OʹHara and of Freedman, He, and Wang. We also find energies not minimized by a round circle. The proof is based on a theorem of Lükő on average chord lengths of closed curves.