Circles as global minimizers of TP^{(p,q)} for the full parameter range q≥1, p∈[q+1,2q+1]

Establish that for every q ≥ 1 and p in the interval [q+1, 2q+1], circles are global minimizers of the generalized tangent-point energy TP^{(p,q)} among all closed, injective curves of fixed length in Euclidean space.

Background

The paper proves a Fenchel-type theorem for Gauss maps and uses it to show that circles minimize generalized tangent-point energies TP{(p,q)} in a broad parameter regime. Specifically, the authors derive sharp lower bounds and prove rigidity/uniqueness for many choices of (p,q), including the geometric case p=2q, the lower limit case p=q+1 (for q>1), and substantial portions around p≈2q; in some cases, the result is shown among convex curves.

Despite these advances, the authors do not cover the entire range p∈[q+1,2q+1] for all q≥1. Motivated by their results at p=q+1 and surrounding ranges, they explicitly conjecture that circles should be global minimizers for the whole interval. Within their framework, they only extend the scope up to ranges governed by a constant c*∈2,2.5, leaving the full interval unresolved.

References

Motivated by our findings concerning $p=q+1$, we conjecture that circles globally minimize $TP{(p,q)}$ for all $q\geq 1, p\in {q+1,2q+1}$.

— A Fenchel Theorem for the Gauss maps and uniqueness of minimizers of nonlocal curvature energies  (2604.02042 - Döhrer et al., 2 Apr 2026) in Subsection “Main Results,” Section 1 (Introduction)