THE REGULAR POLYGON MAXIMISES THE PRODUCT OF DISTANCES OF n POINTS OF GIVEN DIAMETER FOR ODD n≤15

For points z_1, ..., z_n in the plane with diameter at most 2, Erdős, Herzog and Piranian asked for the maximum of the product Δ of all distances |z_i − z_j| (i ≠ j) and whether it is attained by a regular polygon. For even n ≥ 4 it is not. For odd n, Cambie, Decadt, Dong, Hu and Tang conjectured that the regular n-gon is optimal and proved this for n ≤ 5. We prove that for every odd n ≤ 15 the regular n-gon is, up to isometry, the unique maximiser. Using Reinhardt's perimeter bound and Ptolemy inequalities whose fourth side is a chord of length at most 2, we reduce the problem, for every odd n, to one inequality for a function of a few consecutive sides and angles. For n = 7 this inequality is proved by hand; for n = 9, 11 (three variables) and n = 13, 15 (five variables) it is verified by rigorous interval branch-and-bound, each time with two independently written programs. We also prove an a priori angle bound for maximisers, a sharp bound for the product of the two smallest distance classes of convex polygons of diameter at most 2, and strict local maximality of the regular n-gon for all odd n; a natural relaxation of the problem is shown to be false. Source code and certificates are attached: this record contains the manuscript (PDF) and supplementary_code.zip with all source code, logs, sampled certificates and SHA-256 hashes of the full certificates, and a README mapping every statement to the code that verifies it.