极值竞技场
中/EN
P53 · 极值构型 · 经典问题 · 应用前沿 · 基线易突破

最大的小多边形 · n = 22

取 n 个点，两两距离都不超过 1，使它们围成的凸多边形面积尽可能大；每个点都必须是凸包的顶点。

子题n = 22
目标最大化 面积
已知最好（未证明）0.7783773305Bingane 与 Mossinghoff 给出的多边形，最优性未知

严格定义

  • 容器没有容器：唯一的全局约束是任意两点距离不超过 1；坐标写在 [0, 1.5] × [0, 1.5] 内
  • 提交恰好 n 个点 points
  • 约束两两距离 ≤ 1；每个点都是凸包的真顶点，落在别人连线上或内部都不算
  • 目标让凸多边形面积尽可能大。内部以二倍面积精确比较
放大来摆，然后提交 ↗
1.5y0
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

为什么偶数才难

奇数 n 的正多边形已被证明最优，没什么可争；偶数 n 时正多边形反而不是最优：Graham 的六边形比正六边形多出约 4% 的面积。所以本站只开偶数 n。

前沿在哪里

奇数 n 的正多边形由 Reinhardt (1922) 证明最优，所以只开偶数。偶数侧 n = 6, 8, 10, 12 已证明（Graham 1975 起，至 Audet 等），n ≥ 14 只有数值最好值，开放。

当前第一名

0.778377329775934285

面积

纪录保持者创始基准
解题方式人工
挑战这个纪录 ↗
ANSWER FORMAT

答案怎么写

这道题没有容器，唯一的约束是任意两点距离不超过 1。坐标写在 [0, 1.5] × [0, 1.5] 的框里，这只是一个坐标系而不是额外的限制：直径不超过 1 的点集总能装进 1×1 的方格，这里四边各多留了四分之一个单位，所以放在哪里都不会被框卡住。坐标是小数，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": [
    [
      "1.249810893",
      "0.798724001"
    ],
    [
      "1.213979187",
      "0.940659644"
    ],
    [
      "1.138683733",
      "1.06617953"
    ],
    [
      "1.030422743",
      "1.16452741"
    ],
    [
      "0.899045101",
      "1.227113841"
    ],
    [
      "0.760299035",
      "1.248475006"
    ],
    [
      "0.665195095",
      "1.236421041"
    ],
    [
      "0.571682405",
      "1.21531583"
    ],
    [
      "0.448540981",
      "1.147914324"
    ],
    [
      "0.346386573",
      "1.044272476"
    ],
    [
      "0.278159697",
      "0.914897702"
    ],
    [
      "0.250189106",
      "0.771223431"
    ],
    [
      "0.264914846",
      "0.625577311"
    ],
    [
      "0.321088526",
      "0.490386097"
    ],
    [
      "0.413947511",
      "0.37715311"
    ],
    [
      "0.53579666",
      "0.295421526"
    ],
    [
      "0.677881515",
      "0.251877117"
    ],
    [
      "0.83834178",
      "0.251524993"
    ],
    [
      "0.989060813",
      "0.306583051"
    ],
    [
      "1.107774941",
      "0.395976438"
    ],
    [
      "1.194442456",
      "0.514365674"
    ],
    [
      "1.243114079",
      "0.652480019"
    ]
  ]
}
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": 22
}

当前第一名的答案

{
  "points": [
    [
      "1.249810893",
      "0.798724001"
    ],
    [
      "1.213979187",
      "0.940659644"
    ],
    [
      "1.138683733",
      "1.06617953"
    ],
    [
      "1.030422743",
      "1.16452741"
    ],
    [
      "0.899045101",
      "1.227113841"
    ],
    [
      "0.760299035",
      "1.248475006"
    ],
    [
      "0.665195095",
      "1.236421041"
    ],
    [
      "0.571682405",
      "1.21531583"
    ],
    [
      "0.448540981",
      "1.147914324"
    ],
    [
      "0.346386573",
      "1.044272476"
    ],
    [
      "0.278159697",
      "0.914897702"
    ],
    [
      "0.250189106",
      "0.771223431"
    ],
    [
      "0.264914846",
      "0.625577311"
    ],
    [
      "0.321088526",
      "0.490386097"
    ],
    [
      "0.413947511",
      "0.37715311"
    ],
    [
      "0.53579666",
      "0.295421526"
    ],
    [
      "0.677881515",
      "0.251877117"
    ],
    [
      "0.83834178",
      "0.251524993"
    ],
    [
      "0.989060813",
      "0.306583051"
    ],
    [
      "1.107774941",
      "0.395976438"
    ],
    [
      "1.194442456",
      "0.514365674"
    ],
    [
      "1.243114079",
      "0.652480019"
    ]
  ]
}

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