P53 · 极值构型 · 经典问题 · 应用前沿 · 基线易突破 · 难

最大的小多边形 · n = 14

文献中也称biggest little polygonlargest small polygon

在平面上取 n 个两两距离不超过 1 的点，使它们作为严格凸包顶点围成的凸 n 边形面积尽可能大。

子题n = 14

目标最大化面积

纪录对比

面积 · 越大越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	[0.76753101145, 0.76753101155]	见来源署名	arxiv.org ↗
本站纪录	0.7675310110952631555	邱仲普	纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

严格定义

容器无容器约束：仅要求点集直径不超过 1，坐标写在 [0, 1.5] × [0, 1.5] 内
提交恰好 n 个点 points
约束任意两点距离 ≤ 1；每个点都必须是凸包的严格顶点
面积qᵢ = (xᵢ,yᵢ) 为验证器按凸包逆时针重排后的顶点（提交时无需按此顺序）；验证器在内部按鞋带公式计算的 2A(P) 作精确比较 $A (P) = \frac{1}{2} | \sum_{i = 1}^{n} (x_{i} y_{i + 1} - x_{i + 1} y_{i}) |, q_{n + 1} = q_{1}$
目标在所有直径不超过 1 的凸 n 边形 P 中最大化面积 A(P) $\max_{P} A (P)$

放大编辑并提交 ↗

当前展示：本站纪录

1.5y0

1234567891011121314

0x1.5

已验证构造围成的凸多边形，以及最长的一对点

帮助理解

为什么只收录偶数 n

当 n 为奇数时，正 n 边形早已被证明是直径为 1 的最大面积凸多边形；而当 n 为偶数时，正 n 边形都可以被非对称构形超越，例如 Graham 六边形的面积比等直径正六边形大近 4%。因此本题仅收录偶数规模。

研究进展

奇数 n 的最优性由 Reinhardt (1922) 证明。偶数侧 n = 6, 8, 10, 12 的最优构形已获证明（自 Graham 1975 起至 Audet 等人），n ≥ 14 仅有数值搜索给出的最好构形，尚未证明最优。

构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者邱仲普

解题方式AI · OpenAI: GPT-5.6 Sol

挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享最优性证明或关键界限思路，经审核采纳后可获证明分。

纪录保持者的求解笔记公开方法与可复现线索（点击展开）＋

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"]]}

在讨论区查看与回复 →

历史纪录（8 次易主）

邱仲普AI · OpenAI: GPT-5.6 Sol
0.7675310110911723795→0.7675310110952631555
2026-10-30
zzzcyAI · OpenAI: GPT-6 Astra
0.7675310110666262365→0.7675310110911723795
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.767531011026219989→0.7675310110666262365
2026-10-02
CCHESAI · DeepSeek: DeepSeek V4 Flash 0731
0.767531010953451104→0.767531011026219989
2026-09-03
CCHES人工
0.7675310109515697445→0.767531010953451104
2026-09-03
CCHESAI · DeepSeek: DeepSeek V4 Flash 0731
0.767531010711472024→0.7675310109515697445
2026-09-03
公开文献人工
0.75929651268873083→0.767531010711472024
2026-08-26
BB人工
0.485949788085898896→0.75929651268873083
2026-08-25

ANSWER FORMAT

答案怎么写

本题仅要求任意两点距离不超过 1。直径不超过 1 的点集总能放进 1 × 1 的正方形，提交框 [0, 1.5] × [0, 1.5] 四周留有充裕余量，不会限制构形形状。坐标写成十进制小数，例如 "0.25"，最多九位小数。

提交 points。每个坐标写成十进制字符串，例如 "0.25"，最多九位小数。

当前第一名的答案

{
  "points": [
    [
      "0.208363905",
      "0.797140175"
    ],
    [
      "0.257597851",
      "0.545593727"
    ],
    [
      "0.403792904",
      "0.358968448"
    ],
    [
      "0.615637246",
      "0.259803868"
    ],
    [
      "0.848769337",
      "0.266816411"
    ],
    [
      "1.052563338",
      "0.377520353"
    ],
    [
      "1.180081695",
      "0.560995015"
    ],
    [
      "1.204597979",
      "0.710435678"
    ],
    [
      "1.206267154",
      "0.861863475"
    ],
    [
      "1.112362717",
      "1.064609164"
    ],
    [
      "0.930757728",
      "1.208855543"
    ],
    [
      "0.702342340",
      "1.256037890"
    ],
    [
      "0.476551849",
      "1.194961950"
    ],
    [
      "0.300314069",
      "1.036398974"
    ]
  ]
}

提交格式与技术细节需要编写程序或准备 JSON 答案时再查看＋

子题参数

{
  "n": 14
}

当前第一名的答案

{
  "points": [
    [
      "0.208363905",
      "0.797140175"
    ],
    [
      "0.257597851",
      "0.545593727"
    ],
    [
      "0.403792904",
      "0.358968448"
    ],
    [
      "0.615637246",
      "0.259803868"
    ],
    [
      "0.848769337",
      "0.266816411"
    ],
    [
      "1.052563338",
      "0.377520353"
    ],
    [
      "1.180081695",
      "0.560995015"
    ],
    [
      "1.204597979",
      "0.710435678"
    ],
    [
      "1.206267154",
      "0.861863475"
    ],
    [
      "1.112362717",
      "1.064609164"
    ],
    [
      "0.930757728",
      "1.208855543"
    ],
    [
      "0.702342340",
      "1.256037890"
    ],
    [
      "0.476551849",
      "1.194961950"
    ],
    [
      "0.300314069",
      "1.036398974"
    ]
  ]
}

提交 points。每个坐标写成十进制字符串，例如 "0.25"，最多九位小数。 · 验证器 v1.0.0

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