五三角形和七三角形的定量Monsky问题中的精确面积范围最小值
Exact Area-Range Minima in the Quantitative Monsky Problem for Five and Seven Triangles
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中文总结 AI 辅助
本文证明了单位正方形三角形剖分面积范围下确界可达性，并精确求出五、七三角形的极小值，给出九三角形的上界与排除族，核心方法为组合分类与符号证书。
中文摘要 AI 辅助
对于将单位正方形剖分为$n$个非退化三角形的剖分$D$，令$R(D)=\max_i a_i-\min_i a_i$，$\Delta(n)=\inf_D R(D)$。我们证明该下确界对每个$n\ge2$均可达到，并确定了$n=5$和$n=7$时的精确最小值，允许T型连接。对于五个三角形，$\Delta(5)=\frac{5\sqrt5-11}{8}$；等号成立当且仅当三个面积等于$(3-\sqrt5)/4$，两个面积等于$(3\sqrt5-5)/8$。对于七个三角形，$\Delta(7)=r_7$，其中$r_7$是$864r^4+2160r^3-6060r^2+4972r-1$在$(0,1/4900)$内的唯一根。每个极小化剖分有四个面积为$(1+3r_7)/7$，三个面积为$(1-4r_7)/7$，尽管其几何形状不必唯一。证明结合了有限组合分类与精确符号和整数区间证书。对于九个三角形，一个倾斜条带构造给出了显式代数上界$\Delta(9)\le 0.0001273496861283553341\ldots$，这是该拓扑内的精确最小值。反之，在完整的单帽双轨之字形族中，具有任意连续面积的每个剖分的范围都大于$1/3500$；因此全局极小化剖分必须在该族之外。$\Delta(9)$的精确值仍然开放。
英文摘要
For a dissection $D$ of the unit square into $n$ nondegenerate triangles, let $R(D)=\max_i a_i-\min_i a_i, Δ(n)=\inf_D R(D).$ We prove that this infimum is attained for every $n\ge2$, and determine the exact minima for $n=5$ and $n=7$, allowing T-junctions. For five triangles, $Δ(5)=\frac{5\sqrt5-11}{8};$ equality holds precisely when three areas equal $(3-\sqrt5)/4$ and two equal $(3\sqrt5-5)/8$. For seven triangles, $Δ(7)=r_7$, where $r_7$ is the unique root in $(0,1/4900)$ of $864r^4+2160r^3-6060r^2+4972r-1.$ Every minimizer has four areas $(1+3r_7)/7$ and three areas $(1-4r_7)/7$, although its geometry need not be unique. The proofs combine finite combinatorial classification with exact symbolic and integer-interval certificates. For nine triangles, a tilted-strip construction gives the explicit algebraic upper bound $Δ(9)\le 0.0001273496861283553341\ldots,$ which is the exact minimum within that topology. Conversely, every dissection in the complete single-cap two-rail zig-zag family, with arbitrary continuous areas, has range greater than $1/3500$; hence a global minimizer must lie outside that family. The exact value of $Δ(9)$ remains open.
发表机构
- USTC（中国科学技术大学）
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