有限截面法的定量估计

Korovkin定理的非线性算子推广中收敛性的定量估计

Quantitative estimates of convergence in nonlinear operator extensions of Korovkin's theorems

该研究针对实连续函数空间上的弱非线性单调算子序列，给出Korovkin定理非线性版本收敛性的连续模量化定量估计，并附示例阐释理论。

Inequalities for the Gamma Function and estimates for the volume of sections of $B_p^n$

We consider $k$-dimensional central sections of the unit ball of $\ell_p^n$ (denoted $B_p^n$) and we prove that their volume are bounded by the volume of $B_p^n$ whenever $1<p<2$ and $1\le k\le (n-1)/2$ or $k=n-1$. We also consider $0<p<1$ and other cases. We obtain sharp inequalities involving Gamma Function in order to get these results.

Quantitative estimates for nonlinear sampling Kantorovich operators

In this paper, we establish quantitative estimates for nonlinear sampling Kantorovich operators in terms of the modulus of continuity in the setting of Orlicz spaces. This general frame allows us to directly deduce some quantitative estimates of approximation in $L^{p}$-spaces, $1\leq p<\infty $, and in other well-known instances of Orlicz spaces, such as the Zygmung and the exponential spaces. Further, the qualitative order of approximation has been obtained assuming $f$ in suitable Lipschitz classes. The above estimates achieved in the general setting of Orlicz spaces, have been also improve…

Sections of convex bodies in John's and minimal surface area position

We prove several estimates for the volume, mean width, and the value of the Wills functional of sections of convex bodies in John's position, as well as for their polar bodies. These estimates extend some well-known results for convex bodies in John's position to the case of lower-dimensional sections, which had mainly been studied for the cube and the regular simplex. Some estimates for centrally symmetric convex bodies in minimal surface area position are also obtained.

临界集的尖锐定量估计

A sharp quantitative estimate of critical sets

本文为$\mathbb{R}^d$上$\mathcal{C}^1$向量值函数的临界集建立了尖锐的$(d-1)$-Hausdorff测度定量估计，并证明了一般$\mathcal{C}^2$函数（Baire范畴意义下）的临界集具有局部有限测度。

Regular Random Sections of Convex Bodies and the Random Quotient-of-Subspace Theorem

It was shown by G. Pisier that any finite-dimensional normed space admits an $α$-regular $M$-position, guaranteeing not only regular entropy estimates but moreover regular estimates on the diameters of minimal sections of its unit-ball and its dual. We revisit Pisier's argument and show the existence of a \emph{different} position, which guarantees the same estimates for \emph{randomly sampled} sections \emph{with high-probability}. As an application, we obtain a \emph{random} version of V. Milman's Quotient-of-Subspace Theorem, asserting that in the above position, \emph{typical} quotients of…