A concavity inequality for symmetric norms

Convexity or concavity inequalities for Hermitian operators

Matrix versions of some basic convexity inequalities are given. Further results on the same topic are proved in the recent papers on arxiv: 1. Hermitian operators and convex functions, 2. A concavity inequality for symmetric norms, 3. Eigenvalue inequalities for convex and log-convex functions (with Jaspal Singh AUJLA).

$f$-势（Kolmogorov均值）的凸性与凹性

Convexity and concavity of $f$-potentials (Kolmogorov means)

本文给出$f$-势（Kolmogorov均值）凸性与凹性的判别准则，涵盖算术、几何、调和均值及$L^p$范数等特例，并用求积法求出所有满足条件的函数$f$。

Concavity of certain matrix trace and norm functions

We refine Epstein's method to prove joint concavity/convexity of matrix trace functions of the extended Lieb type $Tr{Φ(A^p)^{1/2}Ψ(B^q)Φ(A^p)^{1/2}}^s$, where $Φ$ and $Ψ$ are positive linear maps. By the same method combined with majorization technique, similar properties are proved for symmetric (anti-) norm functions of the form $||{Φ(A^p)σΨ(B^q)}^s||$ involving an operator mean $σ$. Carlen and Lieb's variational method is also used to improve the convexity property of norm functions $||Φ(A^p)^s||$.

Monotonicity versions of Epstein's Concavity Theorem and related inequalities

Many trace inequalities can be expressed either as concavity/convexity theorems or as monotonicity theorems. A classic example is the joint convexity of the quantum relative entropy which is equivalent to the Data Processing Inequality. The latter says that quantum operations can never increase the relative entropy. The monotonicity versions often have many advantages, and often have direct physical application, as in the example just mentioned. Moreover, the monotonicity results are often valid for a larger class of maps than, say, quantum operations (which are completely positive). In this p…

On the Linear Structures Induced by the Four Order Isomorphisms Acting on ${\rm{Cvx}}_0({\mathbb R}^n)$

It is known that the volume functional $\,ϕ\mapsto\int e^{-ϕ}\,$ satisfies certain concavity or convexity inequalities with respect to three of the four linear structures induced by the order isomorphisms acting on ${\rm{Cvx}}_0({\mathbb R}^n)$. In this note we define the fourth linear structure on ${\rm{Cvx}}_0({\mathbb R}^n)$ as the pullback of the standard linear structure under the ${\cal J}$ transform. We show that, interpolating with respect to this linear structure, no concavity or convexity inequalities hold, and prove that a quasi-convexity inequality is violated only by up to a facto…

高斯与凸概率测度的Grünbaum不等式

Grünbaum's inequality for Gaussian and convex probability measures

本文在特定凹性假设下建立了一系列精确的Grünbaum型概率测度不等式并给出等号刻画，推导出高斯测度的Ehrhard-Grünbaum不等式，简化了$s$-凹Radon测度的证明并补全了等号刻画。

Alexandrov-Fenchel不等式中的超凸性超越凸性

Hyperbolicity Beyond Convexity in the Alexandrov-Fenchel Inequality

本文建立了一种广义Alexandrov-Fenchel不等式，通过放宽半正定条件并扩展Alexandrov不等式，推导出面积测度线性泛函的对数凹性和平均截面体内蕴体积的Brunn-Minkowski型不等式，解决了两个公开问题。

Prékopa-Leindler不等式的新无穷小形式及其乘法结构与应用

A new infinitesimal form of the Prékopa-Leindler inequality with multiplicative structure and applications

本文提出一种新的Prékopa-Leindler不等式形式，利用其乘法结构开发替代L_2方法，推导稳定性估计并研究Brunn-Minkowski猜想。

关于反阻塞体之和的体积

On the volume of sums of anti-blocking bodies

研究反阻塞体Minkowski和体积的不等式，证明了Plünnecke-Ruzsa型不等式、V.Milman不等式及Roger-Shephard不等式的反演性，给出了精确常数。

On a convexity property of sections of the cross-polytope

We establish the log-concavity of the volume of central sections of dilations of the cross-polytope (the strong B-inequality for the cross-polytope and Lebesgue measure restricted to an arbitrary subspace).

Gagliardo-Nirenberg不等式与Hölder范数

Gagliardo-Nirenberg inequality with Hölder norms

本研究通过建立连接Lebesgue与Hölder空间的插值引理，将经典Gagliardo-Nirenberg不等式推广至Hölder范数，扩大了适用参数范围。

On a norm inequality for a positive block-matrix

For a positive semidefinite matrix $H= \begin{bmatrix} A&X\\ X^{*}&B \end{bmatrix} $, we consider the norm inequality $ ||H||\leq ||A+B|| $. We show that this inequality holds under certain conditions. Some related topics are also investigated.

Characterisation of $L_p$-norms via Hölder's inequality

We characterise $L_p$-norms on the space of integrable step functions, defined on a probabilistic space, via Hölder's type inequality with an optimality condition.

凹性条件下泛函不等式的定量改进

Quantitative improvements of functional inequalities under concavity properties

该研究在凹性条件下拓展了Barthe与Koldobsky对Hensley泛函界的定量改进，证明了$s$-凹函数的对应上界，推广了泛函形式，得到$p$-范数与熵不等式的定量改进，并讨论其几何与概率意义。

An Inequality of Uniformly Continuous Functions in Normed Spaces

We obtain an interesting inequalities for uniformly continuous functions in the normed spaces: $\|f(x)\|\leq a\|x\|+b$ for some $a,b> 0$.

Variational Representations related to Quantum Rényi Relative Entropies

In this paper, we focus on variational representations of some matrix symmetric norm functions that are related to the quantum Rényi relative entropy. Concretely, we obtain variational representations of the function (A,B)\mapsto \normmm{(B^{q/2}K^*A^pKB^{q/2})^s} for symmetric norms by using the Hölder inequality and Young inequality. These variational expressions enable us to make the proofs of the convexity/concavity of the trace function (A,B)\mapsto \tr (B^{q/2}K^*A^pKB^{q/2})^s more clear.

Additive Reverses of the Generalised Triangle Inequality in Normed Spaces

Some additive reverses of the generalised triangle inequality in normed linear spaces are given. Applications for complex numbers are provided as well.

A matrix subadditivity inequality for symmetric norms

Well-known subadditivity results for positive operators (of Brown-Kosaki and Rotfeld/Ando-Zhan types) are extended to Hermitian and normal ones. Applications to Cartesian decomposition and block-matrices are given.