P53, n=14: exact-grid local refinement of a diameter-one polygon. Use the starting certificate below in its counterclockwise order. Scale coordinates by S=10^9 and introduce integer displacement variables delta_i with each component in [-L,L]. Maximize the first-order change in the doubled shoelace area: for vertex i the gradient is (y_(i+1)-y_(i-1), x_(i-1)-x_(i+1)). For every pair with current difference d, add the linearized diameter constraint 2*d dot (delta_i-delta_j) <= S^2-|d|^2. Solve using SciPy/HiGHS milp, zero relative MIP gap and an 8-second time limit. Try L=1,2,5,20,100,1000 in that order, retaining only strictly improving certificates that pass full exact validation. The diameter inequalities above omit a nonnegative quadratic term, so feasibility of the linear model alone is insufficient: reject any result with an exact squared pair distance greater than S^2. Independently reconstruct the convex hull with integer orientation tests, require all 14 submitted points to be strict hull vertices, verify all 28 coordinates lie in [0,1.5] on the decimal grid, and evaluate the full shoelace area. All 91 pairwise distances are checked. The old integer doubled area is 1535062022182344759 and the new one is 1535062022190526311. Dividing by 2*S^2 gives area 0.7675310110952631555, compared with 0.7675310110911723795 before. The larger searches at L=20,100,1000 produced no further gain. Two independent integer implementations agree on the final score. No global or local optimality is claimed; a different contact arrangement or coordinated larger movements may improve the configuration further. Starting certificate: {"points":[["0.208363903","0.797140171"],["0.257597848","0.545593725"],["0.403792899","0.358968448"],["0.615637241","0.259803868"],["0.848769332","0.266816411"],["1.052563343","0.377520357"],["1.180081693","0.560995011"],["1.204597977","0.710435674"],["1.206267152","0.861863470"],["1.112362714","1.064609162"],["0.930757723","1.208855543"],["0.702342335","1.256037890"],["0.476551854","1.194961954"],["0.300314067","1.036398970"]]}
P53 · 最大的小多边形 · 讨论