论文标题
凸多边形的宽度偏差
Width deviation of convex polygons
论文作者
论文摘要
我们考虑沿随机方向$ n $ n $ -gon $ t $的宽度$ x_t（ω）$沿随机方向$ t $ in \ mathbb {r}/2π\ mathbb {z} $并研究其偏差率：$$ δ（x_t）= \ frac {\ sqrt {\ mathbb {e}（x^2_t） - \ Mathbb {e}（x_t）^2}}} {\ Mathbb {e}（x_t）}。 $$我们证明，当且仅当$ t $变成$ 2 $ -GON时，最大值才能达到最大值。令$ n \ geq 2 $是一个整数，不是$ 2 $的电源。我们表明，$$ \ sqrt {\fracπ{4n \ tan（\fracπ{2n}）} +\ frac {π^2} {8n^2 \ sin^2（\ sin^2（\fracπ{2n}）} - 1} - 1} - 1} $是$ $ $ usumum $ unimim $ unimim $ n $ n $ n $ n $ unem uneus $Δ（x _ t） $ t $达到了这一最低限度。它们的特征是由K.特别是，如果$ n $很奇怪，则常规$ n $ -gon是最低形状之一。当$ n $甚至是$ n $时，我们看到普通的$ n $ gon远非最佳。我们还观察到常规三角形截断的偏差率的意外属性。
We consider the width $X_T(ω)$ of a convex $n$-gon $T$ in the plane along the random direction $ω\in\mathbb{R}/2π\mathbb{Z}$ and study its deviation rate: $$ δ(X_T)=\frac{\sqrt{\mathbb{E}(X^2_T)-\mathbb{E}(X_T)^2}}{\mathbb{E}(X_T)}. $$ We prove that the maximum is attained if and only if $T$ degenerates to a $2$-gon. Let $n\geq 2$ be an integer which is not a power of $2$. We show that $$ \sqrt{\fracπ{4n\tan(\fracπ{2n})} +\frac{π^2}{8n^2\sin^2(\fracπ{2n})}-1} $$ is the minimum of $δ(X_T)$ among all $n$-gons and determine completely the shapes of $T$'s which attain this minimum. They are characterized as polygonal approximations of equi-Reuleaux bodies, found and studied by K.~Reinhardt. In particular, if $n$ is odd, then the regular $n$-gon is one of the minimum shapes. When $n$ is even, we see that regular $n$-gon is far from optimal.We also observe an unexpected property of the deviation rate on the truncation of the regular triangle.
