Tangent-Point Energies and a Fenchel Theorem for the Unit Secant

Self-repulsive energies have become an active field of research during the last years. I will give some basic definitions concerning self-repulsive functionals, and introduce a special family of functionals - the Tangent-Point energies. They are special in the sense that, besides exhibiting self-repulsion, they also provide a notion of nonlocal curvature. After giving a short introduction into this family of energies, I will connect them to other nonlocal curvatures. Subsequently, I will present a Fenchel theorem for the unit secant. After discussing the statement and its proof, I will highlight its connection to the Tangent-Point energies. Ultimately, I will sketch how this can be applied in order to minimize Tangent-Point energies and other nonlocal curvature functionals.