Reinhardt's Maximum-Perimeter Polygon Problem at n=16, 32, and 64: Computer-Assisted Proof Candidates
This addresses a long-standing open problem in convex geometry for the first three open power-of-two cases, providing strong evidence for the conjectured maxima, though the results are preliminary and not yet peer-reviewed.
This paper provides computer-assisted proof candidates for Reinhardt's maximum-perimeter polygon problem at n=16, 32, and 64, asserting uniqueness of the maximizing congruence class in each case. The proofs involve exhaustive exact arithmetic screening of sign codes, leaving 16, 96, and 896 survivors before orbit elimination, respectively, but have not yet been independently verified.
A convex polygon is called small if its diameter is at most one. Reinhardt proved the universal perimeter bound $\mathrm{perim}(P) \leq U_n := 2n\sin(π/(2n))$, and the bound is attained whenever $n$ has a nontrivial odd divisor. The remaining power-of-two cases have resisted exact solution beyond $n=8$. This paper presents computer-assisted proof candidates for the first three open cases, $n=16,32,64$. In each case, the candidate theorem asserts uniqueness of the maximizing congruence class. The proof architecture is common to all three cases: pass to the difference body $P-P$; encode its reconstruction by a sign code; prove that every global maximizer is saturated, so all difference-body vertices lie on the unit circle; localize every competitive configuration near the regular angle vector; exhaustively screen the sign codes using exact arithmetic; eliminate all nonwinning dihedral orbits; and prove uniqueness inside the winning code by strong convexity and a quantitative KKT argument. The exact certificates cover $2^{15}$ normalized codes for $n=16$, $2^{31}$ normalized codes for $n=32$, and all $2^{64}$ half-codes for $n=64$, leaving respectively $16$, $96$, and $896$ survivors before orbit elimination. The accompanying source package contains the verifiers, recorded outputs, and separate computational cross-checks. These results have not yet received independent human expert review and are therefore deliberately presented as proof candidates rather than literature-established theorems.