一个改进的Moser蠕虫问题的下界

莫泽蠕虫问题的平衡支撑校准：三角覆盖的精确证明与边界

Balanced Support Calibrations for Moser's Worm Problem: Exact Certificate and Bound of Triangular Cover

本文针对莫泽蠕虫问题，研究特定等腰三角形的贝尔曼森林迷路逃生路径，通过多类精确分析与穷举分支推导，得到该三角形对应通用覆盖的下界及面积比值。

Viterbo's conjecture as a worm problem

In this paper, we relate Viterbo's conjecture from symplectic geometry to Minkowski versions of worm problems which are inspired by the well-known Moser worm problem from geometry. For the special case of Lagrangian products this relation provides a connection to systolic Minkowski billiard inequalities and Mahler's conjecture from convex geometry. Moreover, we use the above relation in order to transfer Viterbo's conjecture to a conjecture for the longstanding open Wetzel problem which also can be expressed as a systolic Euclidean billiard inequality and for which we discuss an algorithmic ap…

曲线填充和模估计

Curve packing and modulus estimates

如果一个平面曲线族包含长度为1的$\mathbb{R}^{2}$中每一条可求长曲线的等距副本，则该平面曲线族称为Moser族。L. 1966年的Moser要求在任何Moser族中由曲线覆盖的最小面积。1979年，J.M.马斯特兰德证明，答案是不为零：工会的曲线在莫泽家庭始终面积至少$c$为一些小的绝对常数$c > 0$。我们加强Marstrand的结果表明，$p > 3$，$p$ -模的Moser族曲线至少是$c_{p} > 0$。

单位弧的通用凸覆盖的改进界

Improved bounds for universal convex covers of unit arcs

本研究改进 Moser 蠕虫问题的凸通用覆盖面积界至 [0.239, 0.24633]，通过有限细分证明下界和构造四边形证明上界，并用 Lean 4 形式化验证。

Improved upper bounds in the moving sofa problem

The moving sofa problem, posed by L. Moser in 1966, asks for the planar shape of maximal area that can move around a right-angled corner in a hallway of unit width. It is known that a maximal area shape exists, and that its area is at least 2.2195... - the area of an explicit construction found by Gerver in 1992 - and at most $2\sqrt{2}=2.82...$, with the lower bound being conjectured as the true value. We prove a new and improved upper bound of 2.37. The method involves a computer-assisted proof scheme that can be used to rigorously derive further improved upper bounds that converge to the co…