Andreas Kreuml
The minimizers of the anisotropic fractional isoperimetric inequality with respect to the convex body $K$ in $\mathbb{R}^n$ are shown to be equivalent to star bodies whenever $K$ is strictly convex and unconditional. From this a Pólya-Szegö principle for anisotropic fractional seminorms is derived by using symmetrization with respect to star bodies.
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We prove the stability of the ball as global minimizer of an attractive shape functional under volume constraint, by means of mass transportation arguments. The stability exponent is $1/2$ and it is sharp. Moreover, we use such stability result together with the quantitative (possibly fractional) isoperimetric inequality to prove that the ball is a global minimizer of a shape functional involving both an attractive and a repulsive term with a sufficiently large fixed volume and with a suitable (possibly fractional) perimeter penalization.
Topological regularity of isoperimetric sets in PI spaces having a deformation property
本文证明了具有形变性质的PI空间中等周集的拓扑正则性，揭示了其开集、边界密度估计及有界性，并推广至塌缩RCD空间，简化了经典欧几里得论证。
Brunn-Minkowski and Reverse Isoperimetric Inequalities for Dual Quermassintegrals
本文在对偶Brunn-Minkowski理论中建立了两个新几何不等式：证明了Lutwak猜想的原点对称凸体对偶均质积分的Brunn-Minkowski不等式，并推广Ball体积比不等式得出在John位置下立方体最大化对偶均质积分的逆向等周不等式。
Contraction properties for holomorphic functions via isoperimetric stability on the Bergman ball
本文研究全纯函数在近常数情况下的局部收缩性质，通过伯格曼球上的等周稳定性结果，将加权伯格曼空间的缺陷不等式与全纯函数的收缩性质联系起来。
In this note we prove that the ball is a maximiser of some Schatten $p$-norms of the Riesz potential operators among all domains of a given measure in $\mathbb R^{d}$. In particular, the result is valid for the polyharmonic Newton potential operator, which is related to a nonlocal boundary value problem for the poly-Laplacian extending the one considered by M. Kac in the case of the Laplacian, so we obtain and isoperimetric inequalities for its eigenvalues as well, namely, analogues of Rayleigh-Faber-Krahn and Hong-Krahn-Szegö inequalities.
What is the optimal way to cut a convex bounded domain $K$ in Euclidean space $(\mathbb{R}^n,|\cdot|)$ into two halves of equal volume, so that the interface between the two halves has least surface area? A conjecture of Kannan, Lovász and Simonovits asserts that, if one does not mind gaining a universal numerical factor (independent of $n$) in the surface area, one might as well dissect $K$ using a hyperplane. This conjectured essential equivalence between the former non-linear isoperimetric inequality and its latter linear relaxation, has been shown over the last two decades to be of fundame…
We study the existence of Daugavet- and delta-points in the unit sphere of Banach spaces with a $1$-unconditional basis. A norm one element $x$ in a Banach space is a Daugavet-point (resp. delta-point) if every element in the unit ball (resp. $x$ itself) is in the closed convex hull of unit ball elements that are almost at distance $2$ from $x$. A Banach space has the Daugavet property (resp. diametral local diameter two property) if and only if every norm one element is a Daugavet-point (resp. delta-point). It is well-known that a Banach space with the Daugavet property does not have an uncon…
We solve a class of isoperimetric problems on $\mathbb{R}^N $ with respect to weights that are powers of the distance to the origin. For instance we show that if $k\in [0,1]$, then among all smooth sets $Ω$ in $\mathbb{R} ^N$ with fixed Lebesgue measure, $\int_{\partial Ω} |x|^k \, \mathscr{H}_{N-1} (dx)$ achieves its minimum for a ball centered at the origin. Our results also imply a weighted Polya-Szëgo principle. In turn, we establish radiality of optimizers in some Caffarelli-Kohn-Nirenberg inequalities, and we obtain sharp bounds for eigenvalues of some nonlinear problems.
Ball Covering Property on Operators and Calkin Algebra
本文研究了算子上的球覆盖性质与卡林代数，证明了特定条件下卡林代数不满足BCP，并展示了在某些条件下B(X)具有UBCP。
A Banach space with an unconditional basis which is not slicely countably determined
本文研究了一个由无限二叉树分支的适当族生成的巴拿赫空间的单位球几何，并回答了关于切线可数确定巴拿赫空间的相关问题。主要结果是二叉树空间是一个具有无条件基但不切线可数确定的巴拿赫空间，且不包含具有达乌瓦特性质的空间的同构副本。
In this paper we study regularity and topological properties of volume constrained minimizers of quasi-perimeters in $\sf RCD$ spaces where the reference measure is the Hausdorff measure. A quasi-perimeter is a functional given by the sum of the usual perimeter and of a suitable continuous term. In particular, isoperimetric sets are a particular case of our study. We prove that on an ${\sf RCD}(K,N)$ space $({\rm X},{\sf d},\mathcal{H}^N)$, with $K\in\mathbb R$, $N\geq 2$, and a uniform bound from below on the volume of unit balls, volume constrained minimizers of quasi-perimeters are open bou…
Higher-Order Reverse Isoperimetric Inequalities for Log-concave Functions
本文将对数凹函数的 Rogers-Shephard 和 Zhang 投影不等式扩展至 $m$ 阶设置，利用不对称 LYZ 体简化证明，并引入对数凹函数的径向平均体及 Ball 体扩展，提供了泛函推广。
The Orlicz-Brunn-Minkowski theory receives considerable attention recently, and many results in the $L_p$-Brunn-Minkowski theory have been extended to their Orlicz counterparts. The aim of this paper is to develop Orlicz $L_ϕ$ affine and geominimal surface areas for single convex body as well as for multiple convex bodies, which generalize the $L_p$ (mixed) affine and geominimal surface areas -- fundamental concepts in the $L_p$-Brunn-Minkowski theory. Our extensions are different from the general affine surface areas by Ludwig (in Adv. Math. 224 (2010)). Moreover, our definitions for Orlicz $…
The isoperimetric inequality for partial sums of Toeplitz eigenvalues in the Fock space
本文证明在圆对称且测度固定的子集中，球唯一最大化Fock空间上Toeplitz算子前K个特征值之和，并推广至Schatten范数及径向符号，同时发现第二特征值的对称性破缺。
Dual Banach spaces with the ball-covering property
研究对偶巴拿赫空间球覆盖性质及与预对偶单位球几何联系，给出单位球是切片可数确定集的充要条件等，得到对偶空间一致球覆盖性质的充分条件，应用于算子空间和利普希茨空间，回答相关问题。
Approximation properties and quantitative estimation for uniform ball-covering property of operator spaces
本文通过 Schauder 框架膨胀技术推广了 Godefroy-Kalton 逼近定理，刻画了 λ-UBAP 与 1-UFDD 补子空间嵌入的等价关系，并在此基础上给出算子空间 B(X,Y) 具有一致球覆盖性质的充分条件及定量估计。
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