论文标题
石 - 魏尔斯特拉斯定理用于均质多项式及其在凸几何中的作用
Stone-Weierstrass theorem for homogeneous polynomials and its role in convex geometry
论文作者
论文摘要
We give a uniform approximation of the characteristic function of the boundary of a centrally symmetric n-dimensional compact and convex set by homogeneous polynomials of even degree $d$ fulfilling $|g_d-1|\leq E/d^{1/2-β}$, for every $β>0$, large enough $d$, and some constant $E$ only depending on $n$ and $K$.特别是，这证明了Kroo在2004年提出的一种猜想，也称为统一多项式的Stone-Weiersstrass定理。 此外，我们通过其d-lasserre-löwner多项式介绍了凸件$ k $ in $ \ mathbb r^n $的D-VOLUME比率。我们还证明了$ 1+f/d^{3/2-β} $的d-volume比率的上限，对于每$β> 0 $，足够大的$ d $和$ f $的上限，仅取决于$ n $。
We give a uniform approximation of the characteristic function of the boundary of a centrally symmetric n-dimensional compact and convex set by homogeneous polynomials of even degree $d$ fulfilling $|g_d-1|\leq E/d^{1/2-β}$, for every $β>0$, large enough $d$, and some constant $E$ only depending on $n$ and $K$. In particular, this proves a conjecture posed by Kroo in 2004, also known as the Stone-Weierstrass theorem for homogeneous polynomials. Moreover, we introduce the d-volume ratio for a convex body $K$ in $\mathbb R^n$, by means of its d-Lasserre-Löwner polynomial. We also prove an upper bound of the d-volume ratio of the form $1+F/d^{3/2-β}$, for every $β>0$, large enough $d$, and $F$ some constant only depending on $n$.
