Let $V$ and $W$ be any convex and origin-symmetric bodies in ${\mathbb{R}}^n$ . Assume that for some $\mathbf{A\; \in }\mathcal{L}\left({\mathbb{R}}^n\to {\mathbb{R}}^n\right)$, $det\mathbf{A}\ne 0$, $V$ is contained in the ellipsoid ${\mathbf{A}}^{-1}{B}_{\left(2\right)}^n$, where $B_{\left(2\right)}^n$ is the unit Euclidean ball. We give a lower bound for the $W$-radius of sections of ${\mathbf{A}}^{-1}V$ in terms of the spectral radius of ${\mathbf{A}}^{\ast }\mathbf{A}$ and the expectations of ${\Vert \cdot \Vert }_V$ and ${\Vert \cdot \Vert }_{W^o}$ with respect to Haar measure on ${\mathbb{S}}^{n-1}\subset {\mathbb{R}}^n$. It is shown that the respective expectations are bounded as $n\to \infty$ in many important cases. As an application we offer a new method of evaluation of $n$-widths of multiplier operators. As an example we establish sharp orders of $n$-widths of multiplier operators $\Lambda :L_p\left({\mathbb{M}}^d\right)\to L_q\left({\mathbb{M}}^d\right)$, $1<q\le 2\le p<\infty$ on compact homogeneous Riemannian manifolds ${\mathbb{M}}^d$. Also, we apply these results to prove the existence of flat polynomials on ${\mathbb{M}}^d$.

让 $V$ 和 $\mathrm{w\; ^}$ 是任何凸且原点对称的物体 ${\mathbb{[R}}^{ñ}$。假设$\mathbf{\in }\mathcal{大号}\left({\mathbb{[R}}^{ñ}\to {\mathbb{[R}}^{ñ}\right)$， $det\mathbf{一种}\ne 0$， $V$ 包含在椭球中 ${\mathbf{一种}}^{-\mathrm{1个}}{乙}_{\left(2\right)}^{ñ}$，在哪里 ${乙}_{\left(2\right)}^{ñ}$是单位欧几里得球。我们给$\mathrm{w\; ^}$-部分的半径 ${\mathbf{一种}}^{-\mathrm{1个}}V$ 根据光谱半径 ${\mathbf{一种}}^{\ast }\mathbf{一种}$ 和期望 ${\Vert \cdot \Vert }_V$ 和 ${\Vert \cdot \Vert }_{{\mathrm{w\; ^}}^{Ø}}$ 关于Haar措施 ${\mathbb{小号}}^{ñ-\mathrm{1个}}\subset {\mathbb{[R}}^{ñ}$。结果表明，各自的期望被限制为$ñ\to \infty$在许多重要情况下。作为应用程序，我们提供了一种新的评估方法$ñ$乘运算符的宽度。例如，我们建立了$ñ$运算符的宽度 $\Lambda ：{\mathrm{大号}}_p\left({\mathbb{中号}}^d\right)\to {\mathrm{大号}}_q\left({\mathbb{中号}}^d\right)$， $\mathrm{1个}<q\le 2\le p<\infty$ 在紧齐齐的黎曼流形上 ${\mathbb{中号}}^d$。此外，我们将这些结果应用于证明平多项式存在${\mathbb{中号}}^d$。