Ron Kerman, Rastislav Ol'hava, Susanna Spektor
We solve the open problem of determining the second order term in the asymptotic expansion of the integral in Ball's integral inequality. In fact, we provide a method by which one can compute any term in the expansion. We also indicate how to derive an asymptotically sharp form of a generalized Ball's integral inequality.
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From Clarkson-McCarthy inequality to Ball-Carlen-Lieb inequality
本文利用酉轨道技术、Hadamard 三线定理与对偶论证，推广 Clarkson-McCarthy 不等式并完善 Ball-Carlen-Lieb 的最优2-一致凸性不等式。
To prove by probabilistic methods that every $(n-1)$-dimensional section of the unit cube in $R^n$ has volume at most $\sqrt 2$, K. Ball made essential use of the inequality $$ \frac{1}π\int_{-\infty}^{\infty} \left(\frac{\sin^2 t}{t^2}\right)^pdt\leq \frac{\sqrt 2}{\sqrt p}, \quad p\geq 1, $$ in which equality holds if and only if $p=1$. The right side of above inequality has the correct rate of decay though the limit of the ratio of the right to left side is ${\sqrt{\frac{3}π}}$ rather then $\sqrt 2$. Applying Ball's methods we put all of this into the improved form of the Ball's inequality.
In [18] we have shown that, for $p_{1},p_{2}\in(2,\infty]$, the constants of Bennett's inequality on unimodular bilinear forms on $\ell_{p_{1}}^{n_{1} }\times\ell_{p_{2}}^{n_{2}}$ are asymptotically bounded by $1$. In the present paper we use a different approximation technique to investigate the remaining cases $p_{1},p_{2}\in\lbrack1,\infty].$ This new approach also provides a stronger asymptotic control, in the sense that the constants are "uniformly" asymptotically bounded by $1$, with no dependence on $p_{1},p_{2}.$
We establish a sharp moment comparison inequality between an arbitrary negative moment and the second moment for sums of independent uniform random variables, which extends Ball's cube slicing inequality.
We prove fractional Hardy--Sobolev--Maz'ya inequality for balls and a half-space, partially answering the open problem posed by Frank and Seiringer [arXiv:0906.1561v1 [math.FA], 2009] We note that for half-spaces this inequality has been recently obtained by Sloane [arXiv:1004.4828v1 [math.FA], 2010].
Let $K$ be a convex body and $K^\circ$ its polar body. Call $ϕ(K)=\frac{1}{|K||K^\circ|}\int_K\int_{K^\circ}< x,y>^2 dxdy$. It is conjectured that $ϕ(K)$ is maximum when $K$ is the euclidean ball. In particular this statement implies the Blaschke-Santalo inequality. We verify this conjecture when $K$ is restricted to be a $p$--ball.
We prove a quantitative form of the celebrated Ball's theorem on cube slicing in $\mathbb{R}^n$ and obtain, as a consequence, equality cases in the min-entropy power inequality. Independently, we also give a quantitative form of Khintchine's inequality in the special case $p=1$.
The best constant of the Sobolev inequality in the whole space is attained by the Aubin-Talenti function; however, this does not happen in bounded domains because the break in dilation invariance. In this paper, we investigate a new scale invariant form of the Sobolev inequality in a ball and show that its best constant is attained by functions of the Aubin-Talenti type. Generalization to the Caffarelli-Kohn-Nirenberg inequality in a ball is also discussed.
We study several aspects concerning slice regular functions mapping the quaternionic open unit ball into itself. We characterize these functions in terms of their Taylor coefficients at the origin and identify them as contractive multipliers of the Hardy space. In addition, we formulate and solve the Nevanlinna-Pick interpolation problem in the class of such functions presenting necessary and sufficient conditions for the existence and for the uniqueness of a solution. Finally, we describe all solutions to the problem in the indeterminate case.
On the Three Balls Inequality for Discrete Schr{ö}dinger Operators on Certain Periodic Graphs
本文利用 Carleman 估计研究周期图上离散磁性 Schrödinger 算子的定量唯一延拓，并将三球不等式推广至包含六角晶格的周期图族。
von Neumann Inequality for a class of Doubly Contractive Weighted Shifts
本文研究了双合同加权位移的球版本von Neumann不等式，证明了平衡或满足适当权重条件的加权位移具有球单位正交扩张，从而满足欧几里得单位球上的von Neumann不等式，并进一步建立了一般双合同算子元的多项式不等式。
We re-confirm, for the case of the unit p-ball of R^n, one of recent conjectures of G.Kuperberg on centrally symmetric convex bodies.This conjecture was very recently confirmrd for this particular case by D.A.Gutierrez using polygamma functions and convexity theory.We present another proof using only the basic properties of gamma function and mildly advanced classical analysis tools.
We obtain all extreme and exposed points of the closed unit ball of the space of bilinear forms $T:\ell_{\infty}^{2}\times\ell_{\infty}^{2}\rightarrow \mathbb{R}.$ We also show that any (norm one) bilinear form $T:\ell_{\infty }^{2}\times\ell_{\infty}^{2}\rightarrow\mathbb{R}$ for which the optimal constant of the Littlewood's $4/3$ inequality is achieved is necessarily an extreme point. In the case of complex scalars we combine analytical and numerical evidences supporting that, at least for complex bilinear forms with real coefficients, the optimal constant of Littlewood's $4/3$ inequality s…
We consider a homogeneous space $X=(X,d,m) $ of dimension $ν\geq1$ and a local regular Dirichlet form in $L^{2}(X,m) .$ We prove that if a Poincaré inequality holds on every pseudo-ball $B(x,R) $ of $X$, then an Harnack's inequality can be proved on the same ball with local characteristic constant $c_{0}$ and $c_{1}$
The Euclidean concentration inequality states that, among sets with fixed volume, balls have $r$-neighborhoods of minimal volume for every $r>0$. On an arbitrary set, the deviation of this volume growth from that of a ball is shown to control the square of the volume of the symmetric difference between the set and a ball. This sharp result is strictly related to the physically significant problem of understanding near maximizers in the Riesz rearrangement inequality with a strictly decreasing radially decreasing kernel. Moreover, it implies as a particular case the sharp quantitative Euclidean…
A direct approach to Ball's simplex inequality is presented. This approach, which does not use the Brascamp-Lieb inequality, also gives Barthe's characterization of the simplex for Ball's inequality and extends it from discrete to arbitrary measures. It also yields the dual inequality, along with equality conditions, and it does both for arbitrary measures.
Convergence analysis for pseudo-monotone variational inequality problem involving projections onto a moving ball
针对伪单调变分不等式问题传统投影法的局限，提出一种向可行集内显式表达的球投影、无需预知代价算子Lipschitz常数的迭代格式，并通过数值实验验证了其有效性。
Stability versions of the functional forms of the Blaschke-Santalo inequality due to Ball, Artstein-Klartag-Milman, Fradelizi-Meyer and Lehec are proved.
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.
We investigate the validity and the stability of various Minkowski-like inequalities for $C^1$-perturbations of the ball. Let $K\subseteq\mathbb R^n$ be a domain (possibly not convex and not mean-convex) which is $C^1$-close to a ball. We prove the sharp geometric inequality $$ \left(\int_{\partial K} \lVert II\rVert_1 d\mathscr H^{n-1}\right)^{\frac1{n-2}} \ge C_1(n)Per(K)^{\frac1{n-1}} , $$ where $C_1(n)$ is the constant that yields the equality when $K=B_1$ (and $\lVert II\rVert_1$ is the sum of the absolute values of the eigenvalues of the second fundamental form $II$ of $\partial K$). Mor…
We establish new functional versions of the Blaschke-Santaló inequality on the volume product of a convex body which generalize to the non-symmetric setting an inequality of K. Ball and we give a simple proof of the case of equality. As a corollary, we get some inequalities for $\log$-concave functions and Legendre transforms which extend the recent result of Artstein, Klartag and Milman, with its equality case.
Distributional stability of the Szarek and Ball inequalities
本文证明 Szarek 最优 Khinchin 不等式与 Ball 立方体切片不等式在接近 Rademacher 分布且权重受 1/√2 分数约束时的分布稳定性，采用傅里叶分析方法。
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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