论文标题
最大周长和最大宽度的凸面小宽度的紧密界限
Tight bounds on the maximal perimeter and the maximal width of convex small polygons
论文作者
论文摘要
小多边形是单位直径的多边形。当$ s \ ge 4 $时，尚不清楚带有$ n = 2^s $顶点的凸小多边形的最大周长和最大宽度。在本文中，我们构建了一个小型$ n $ gons的家族，$ n = 2^s $和$ s \ ge 3 $，并表明，对于某些积极的常数$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ n $ and $ n $ a $ and $ n $的范围和宽度不能分别提高超过$ a/n^6 $和$ a/n^6 $和$ b/n^4 $。此外，假设Mossinghoff的猜想是正确的，我们将最大周围问题作为一个非线性优化问题，涉及三角函数，对于$ n = 2^s $，带有$ 3 \ le s \ le S \ le 7 $，我们提供了全局最佳解决方案。
A small polygon is a polygon of unit diameter. The maximal perimeter and the maximal width of a convex small polygon with $n=2^s$ vertices are not known when $s \ge 4$. In this paper, we construct a family of convex small $n$-gons, $n=2^s$ and $s\ge 3$, and show that the perimeters and the widths obtained cannot be improved for large $n$ by more than $a/n^6$ and $b/n^4$ respectively, for certain positive constants $a$ and $b$. In addition, assuming that a conjecture of Mossinghoff is true, we formulate the maximal perimeter problem as a nonlinear optimization problem involving trigonometric functions and, for $n=2^s$ with $3 \le s\le 7$, we provide global optimal solutions.
