凸小多边形的最大周长和最大宽度的紧界
Tight bounds on the maximal perimeter and the maximal width of convex small polygons
摘要：
小多边形是单位直径的多边形。当满足$s \ge 4$条件时，顶点为$n=2^s$的凸小多边形的最大周长和最大宽度是未知的。本文构造了一类凸小$n$ -簇，$n=2^s$和$s\ge 3$，并证明了对于某些正的常数$a$和$b$，对于大的$n$，所得到的周长和宽度的改进不能分别超过$a/n^6$和$b/n^4$.此外，假设一个猜想的Mossinghoff是真的，我们制定的最大周长问题作为一个非线性优化问题，涉及三角函数，并为$n=2^s$与$3 \le s\le 7$，我们提供了全球最优解。
英文摘要：
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.
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