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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 = 24

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 = 24
目标最大化 面积
已知最好（未证明）0.7795240455Bingane 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

0.779524044705923001

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": [
    [
      "1.249730591",
      "0.798810144"
    ],
    [
      "1.218697877",
      "0.929085326"
    ],
    [
      "1.154253075",
      "1.046523387"
    ],
    [
      "1.060824116",
      "1.142880157"
    ],
    [
      "0.943707707",
      "1.211882403"
    ],
    [
      "0.801978586",
      "1.249955249"
    ],
    [
      "0.655489366",
      "1.241145975"
    ],
    [
      "0.526894014",
      "1.197091499"
    ],
    [
      "0.416004627",
      "1.121479708"
    ],
    [
      "0.329270721",
      "1.019391232"
    ],
    [
      "0.27268686",
      "0.898011971"
    ],
    [
      "0.250269409",
      "0.765987257"
    ],
    [
      "0.26361621",
      "0.632742605"
    ],
    [
      "0.311768017",
      "0.507803676"
    ],
    [
      "0.391227139",
      "0.400155519"
    ],
    [
      "0.495817203",
      "0.317793987"
    ],
    [
      "0.614109094",
      "0.267761249"
    ],
    [
      "0.700965243",
      "0.255070178"
    ],
    [
      "0.78859971",
      "0.250044751"
    ],
    [
      "0.914533826",
      "0.275280577"
    ],
    [
      "1.033543605",
      "0.334939429"
    ],
    [
      "1.133017732",
      "0.424420021"
    ],
    [
      "1.205298777",
      "0.53713109"
    ],
    [
      "1.245154483",
      "0.664973944"
    ]
  ]
}
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": 24
}

当前第一名的答案

{
  "points": [
    [
      "1.249730591",
      "0.798810144"
    ],
    [
      "1.218697877",
      "0.929085326"
    ],
    [
      "1.154253075",
      "1.046523387"
    ],
    [
      "1.060824116",
      "1.142880157"
    ],
    [
      "0.943707707",
      "1.211882403"
    ],
    [
      "0.801978586",
      "1.249955249"
    ],
    [
      "0.655489366",
      "1.241145975"
    ],
    [
      "0.526894014",
      "1.197091499"
    ],
    [
      "0.416004627",
      "1.121479708"
    ],
    [
      "0.329270721",
      "1.019391232"
    ],
    [
      "0.27268686",
      "0.898011971"
    ],
    [
      "0.250269409",
      "0.765987257"
    ],
    [
      "0.26361621",
      "0.632742605"
    ],
    [
      "0.311768017",
      "0.507803676"
    ],
    [
      "0.391227139",
      "0.400155519"
    ],
    [
      "0.495817203",
      "0.317793987"
    ],
    [
      "0.614109094",
      "0.267761249"
    ],
    [
      "0.700965243",
      "0.255070178"
    ],
    [
      "0.78859971",
      "0.250044751"
    ],
    [
      "0.914533826",
      "0.275280577"
    ],
    [
      "1.033543605",
      "0.334939429"
    ],
    [
      "1.133017732",
      "0.424420021"
    ],
    [
      "1.205298777",
      "0.53713109"
    ],
    [
      "1.245154483",
      "0.664973944"
    ]
  ]
}

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