AI 中文总结
该研究证明圆弧$A_\alpha$为$d$-多项式凸的充要条件，探讨光滑Jordan弧的$d$-多项式凸性与总绝对曲率的关系，构造反例并确定$d=2$时的尖锐阈值。
AI 中文摘要
我们证明圆弧$A_\alpha=\{e^{it};|t|\le\alpha\}$是$d$-多项式凸的当且仅当$\alpha\le\frac{d-1}{d}\pi$。随后研究光滑Jordan弧的$d$-多项式凸性与其总绝对曲率$T(K)$的关系：对任意$\tau>\frac{d-1}{d}\pi$，构造了总绝对曲率$T(K)<\tau$但非$d$-多项式凸的光滑Jordan弧$K$；当$d=2$时该阈值是尖锐的，即$T(K)\le\pi/2$意味着$K$是2-多项式凸的。
英文摘要
Let $d\ge 1$ be an integer. We study $d$-polynomial convexity of smooth Jordan arcs in terms of their total absolute curvature $\T(K)$. We prove that every $\cC^2$-smooth Jordan arc $K\subset\C$ satisfying $\T(K)\le \frac{d-1}{d}π$ is $d$-polynomially convex. This bound is sharp: for every $τ>\frac{d-1}{d}π$, there exists a smooth Jordan arc $K\subset\C$ such that $\T(K)<τ$ and $K$ is not $d$-polynomially convex. We also show that, for $0<α<π$, the circular arc $A_α=\{e^{it}:|t|\le α\}$ is $d$-polynomially convex if and only if $α\le \frac{d-1}{d}π$.
Comments19 pages
