关于C $^2$ -光滑等宽曲面
On C$^2$-smooth Surfaces of Constant Width
摘要：
得到了欧氏空间${\mathbb{E}}^3$中C-$^2$-光滑常宽曲面的若干结果.特别地，建立了常宽曲面的一个积分不等式。这是用来证明体积的比例，以立方宽度的一个恒定的宽度表面是减少收缩它沿其法线。我们还给出了具有有理支撑函数的常宽曲面的一个刻划。我们的技术，这是复杂的微分几何的性质，使我们能够构建明确的光滑表面的恒定宽度的${\mathbb{E}}^3$，和他们的焦点集。它们还允许容易地构造具有恒定宽度的四面体对称表面。
英文摘要：
A number of results for C$^2$-smooth surfaces of constant width in Euclidean 3-space ${\mathbb{E}}^3$ are obtained. In particular, an integral inequality for constant width surfaces is established. This is used to prove that the ratio of volume to cubed width of a constant width surface is reduced by shrinking it along its normal lines. We also give a characterization of surfaces of constant width that have rational support function. Our techniques, which are complex differential geometric in nature, allow us to construct explicit smooth surfaces of constant width in ${\mathbb{E}}^3$, and their focal sets. They also allow for easy construction of tetrahedrally symmetric surfaces of constant width.
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