局部代数可积体的新例子
New examples of locally algebraically integrable bodies
摘要：
任何具有光滑边界的${\mathbb R}^N$中的紧致体在仿射超平面空间上定义了一个二值函数：这些超平面将物体切割成两部分的体积。如果$N$是偶数，这个函数就绝不是代数的;如果$N$是奇数，这个函数就很少是代数的：所有已知的定义代数体积函数的物体的例子（基本上是由阿基米德\cite{arch}为$N=3$发现的）都是椭圆体。我们证明了一个大的新系列的{ \em本地}代数可积体的代数边界的空间中的任意尺寸，也就是说，机构，使相应的体积函数符合代数的至少在某些开放域的超平面空间相交的身体。此外，我们讨论（也遵循阿基米德）这个问题的扩展到非紧机构
英文摘要：
Any compact body in ${\mathbb R}^N$ with smooth boundary defines a two-valued function on the space of affine hyperplanes: the volumes of two parts into which these hyperplanes cut the body. This function is never algebraic if $N$ is even and is very rarely algebraic if $N$ is odd: all known examples (found essentially by Archimedes \cite{arch} for $N=3$) of bodies defining algebraic volume functions are these of ellipsoids. We demonstrate a large new series of {\em locally} algebraically integrable bodies with algebraic boundaries in the spaces of arbitrary dimensions, that is, of bodies such that the corresponding volume functions coincide with algebraic ones at least in some open domains of the space of hyperplanes intersecting the body. Further, we discuss (also following Archimedes) the extension of this problem to non-compact bodies
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