AI 中文总结
本文通过森林逃逸问题的最小直径对偶，将勒贝格通用覆盖问题转化为优化问题，并给出离散化与严格认证的数值方法。
AI 中文摘要
勒贝格通用覆盖问题寻求面积最小的凸平面区域，使其能够包含直径至多为一的每个平面集合的一个全等副本。本文通过贝尔曼森林迷路问题的最小直径类比，发展了该问题的精确对偶表述。对于紧凸森林$F$，我们将临界逃逸直径$D(F)$定义为一条封闭路径的最小直径，该路径的轨迹在任何刚体运动下都不能完全置于$F$的内部。我们证明了该最小值的可达性，并建立了直径覆盖/逃逸对偶性，表明归一化体$D(F)^{-1}F$是勒贝格通用覆盖。因此，勒贝格通用覆盖常数具有精确表示\\[ \mathcal L=\inf_F\frac{\operatorname{Area}(F)}{D(F)^2}，\\]其中下确界遍及具有非空内部的紧凸平面体。通过反转刚体运动，我们进一步将逃逸刻画为固定路径与$F$的每个反向变换边界的交集，并推导出等价的连续曲线、凸体和支持函数优化表述。为了使无限维问题在计算上可行，同时保持对近似误差的严格控制，我们通过$\eta_m$-网对紧配置空间进行离散化，并将所得问题表述为具有邻域的最小直径旅行商问题；对于多边形森林，获得了精确的混合整数二阶锥表述。我们证明了定量认证，从而产生收敛的、严格认证的通用覆盖界。该框架用对整个配置空间的统一优化取代了对规定常宽形状的有限测试，并自然扩展到其他全等和平移通用覆盖问题。
英文摘要
Lebesgue's universal covering problem asks for the minimum area convex planar region capable of containing a congruent copy of every planar set of diameter at most one. In this paper, we develop an exact dual formulation of this problem through a minimum diameter analogue of Bellman's lost-in-a-forest problem. For a compact convex forest $F$, we define the critical escape diameter $D(F)$ as the minimum diameter of a closed path whose trace cannot be placed, under any rigid motion, entirely in the interior of $F$. We prove attainment of this minimum and establish a diameter cover/escape duality showing that the normalized body $D(F)^{-1}F$ is a Lebesgue universal cover. Consequently, the Lebesgue universal covering constant admits the exact representation \[ \mathcal L=\inf_F\frac{\operatorname{Area}(F)}{D(F)^2}, \] where the infimum ranges over compact convex planar bodies with nonempty interior. By reversing the rigid motion, we further characterize escape as intersection of a fixed path with every oppositely transformed boundary of $F$, and derive equivalent continuous curve, convex body, and support function optimization formulations. To make the infinite dimensional problem computationally tractable while retaining rigorous control of approximation error, we discretize the compact configuration space by an $η_m$-net and formulate the resulting problem as a minimum diameter traveling salesman problem with neighborhoods; for polygonal forests, an exact mixed-integer second-order cone formulation is obtained. We prove the quantitative certification, which yields convergent, rigorously certified universal cover bounds. The framework replaces finite tests of prescribed constant width shapes by a unified optimization over the full configuration space and extends naturally to other congruent and translative universal cover problems.
Comments49 pages, with Lean 4 code for formal proofs
