论文标题
均匀空间中多项式整合体的类似物
An analog of polynomially integrable bodies in even-dimensional spaces
论文作者
论文摘要
如果$（n-1）$ - 与超平面$π$多条件上的$（n-1）$ - 尺寸$ k $多发的距离，则有限的域$ k \ subset \ subset \ mathbb r^n $可称为多项式集成。在[7]中证明，如果$ n $均匀，则没有这样的域，如果$ n $是奇数，那么唯一具有光滑边界的多项式集成域是椭圆形的。在本文中，我们将多项式可合转性的概念偶数为$ n $，并考虑了截面体积函数是多项式的主体，直到一个因子是二次多项式的平方根，或者等效地，此函数的希尔伯特变换是多项式的。我们证明，均匀的椭圆形是唯一满足该特性的无限光滑身体。
A bounded domain $K \subset \mathbb R^n$ is called polynomially integrable if the $(n-1)$-dimensional volume of the intersection $K$ with a hyperplane $Π$ polynomially depends on the distance from $Π$ to the origin. It was proved in [7] that there are no such domains with smooth boundary if $n$ is even, and if $n$ is odd then the only polynomially integrable domains with smooth boundary are ellipsoids. In this article, we modify the notion of polynomial integrability for even $n$ and consider bodies for which the sectional volume function is a polynomial up to a factor which is the square root of a quadratic polynomial, or, equivalently, the Hilbert transform of this function is a polynomial. We prove that ellipsoids in even dimensions are the only convex infinitely smooth bodies satisfying this property.
