三维中与极直径图相关的等周问题：理论与数值方面
Isoperimetric problems related to extremal diameter graphs in 3D: theoretical and numerical aspects
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中文总结 AI 辅助
该研究针对三维极单位直径构型的对偶边长度可分离泛函优化问题，通过理论分析与数值实验，发现相关下界由正四面体达到，且数值结果为Blaschke-Lebesgue问题的结构猜想提供了支撑。
中文摘要 AI 辅助
我们研究三维有限极单位直径构型中对偶边对的欧氏长度或球面长度的可分离泛函的优化问题。对于固定的直径图，这些问题会导致对顶点坐标的非凸约束优化。我们首先证明，当重合点被合并且关联至多一条直径的顶点被移除后，收敛的极构型序列会保留一个极几何核心。随后，球面Crofton论证为加性凹泛函提供了一个尖锐下界，该下界由正四面体达到。我们还分析了插入或删除悬垂顶点的影响。对于球面对偶边长度的乘积和，我们得到了三维Blaschke-Lebesgue面积问题的精确上确界重构。数值研究使用了所有10644个可用的顶点数不超过16的极构型，我们结合直接评估与每个固定图上基于梯度的局部优化，并在顶点碰撞后重构内在直径图及其对偶对。对于每个提供的图，选定的验证端点包含一个正四面体。这些计算未对任何固定图的全局最大值进行认证，但它们激发了将四面体包含性与Blaschke-Lebesgue问题关联起来的结构猜想。
英文摘要
We study optimization problems for separable functionals of the Euclidean or spherical lengths of dual edge pairs in finite extremal unit-diameter configurations in three dimensions. For a fixed diameter graph, these problems lead to nonconvex constrained optimization of the vertex coordinates. We first prove that a convergent sequence of extremal configurations retains an extremal geometric core after coincident points are merged and vertices incident to at most one diameter are removed. A spherical Crofton argument then gives a sharp lower bound for additive concave functionals, attained by the regular tetrahedron. We also analyze the effect of inserting or deleting dangling vertices. For the sum of products of spherical dual-edge lengths, we obtain an exact supremal reformulation of the three-dimensional Blaschke--Lebesgue area problem. The numerical study uses all 10,644 available extremal configurations with at most 16 vertices. We combine direct evaluation with gradient-based local optimization on each fixed graph and reconstruct the intrinsic diameter graph and its dual pairs after vertex collisions. For every supplied graph, the selected verified endpoint contains a regular tetrahedron. These computations do not certify the global maximum for any fixed graph, but they motivate structural conjectures connecting tetrahedral containment with the Blaschke--Lebesgue problem.
