Related papers: Extensions of Brunn-Minkovski's inequality to mult…

We first present a determinant inequality related to partial traces for positive semidefinite block matrices. Our result extends a result of Lin [Czech. Math. J. 66 (2016)] and improves a result of Kuai [Linear Multilinear Algebra 66…

In this paper, we present some extensions of the Young and Heinz inequalities for the Hilbert-Schmidt norm as well as any unitarily invariant norm. Furthermore, we give some inequalities dealing with matrices. More precisely, for two…

The Brunn-Minkowski inequality states that for bounded measurable sets $A$ and $B$ in $\mathbb{R}^n$, we have $|A+B|^{1/n} \geq |A|^{1/n}+|B|^{1/n}$. Also, equality holds if and only if $A$ and $B$ are convex and homothetic sets in…

The classical Brunn-Minkowski inequality states that for $A_1,A_2\subset\R^n$ compact, $$ |A_1+A_2|^{1/n}\ge |A_1|^{1/n}+|A_2|^{1/n}\eqno(1) $$ where $|\cdot|$ denotes the Lebesgue measure on $\R^n$. In 1986 V. Milman {\bf [Mil 1]}…

A generalization of Young's inequality for convolution with sharp constant is conjectured for scenarios where more than two functions are being convolved, and it is proven for certain parameter ranges. The conjecture would provide a unified…

About last 70s, Haynsworth [6] used a result of the Schur complement to refine a determinant inequality for positive definite matrices. Haynsworth's result was improved by Hartfiel [5]. We extend their result to a larger class of matrices,…

We prove a sharp stability result for the Brunn-Minkowski inequality for $A,B\subset\mathbb{R}^2$. Assuming that the Brunn-Minkowski deficit $\delta=|A+B|^{\frac{1}{2}}/(|A|^\frac12+|B|^\frac12)-1$ is sufficiently small in terms of…

In this paper we present new versions of the classical Brunn-Minkowski inequality for different classes of measures and sets. We show that the inequality \[ \mu(\lambda A + (1-\lambda)B)^{1/n} \geq \lambda \mu(A)^{1/n} +…

Let $A$ and $ B$ be $n\times n$ positive definite complex matrices, let $\sigma$ be a matrix mean, and let $f : [0,\infty)\to [0,\infty)$ be a differentiable convex function with $f(0)=0$. We prove that $$f^{\prime}(0)(A \sigma B)\leq…

A quantitative version of Minkowski sum, extending the definition of $\theta$-convolution of convex bodies, is studied to obtain extensions of the Brunn-Minkowski and Zhang inequalities, as well as, other interesting properties on Convex…

We prove that for positive semidefinite matrices $A$ and $B$ the following determinantal inequality holds: \[ \det(I+A\#B)\le \det(I+A^{1/2}B^{1/2}), \] where $A\#B$ is the geometric mean of $A$ and $B$. We apply this inequality to the…

Hadamard's determinant inequality was refined and generalized by Zhang and Yang in [Acta Math. Appl. Sinica 20 (1997) 269-274]. Some special cases of the result were rediscovered recently by Rozanski, Witula and Hetmaniok in [Linear Algebra…

The Brunn-Minkowski Theorem asserts that $\mu_d(A+B)^{1/d}\geq \mu_d(A)^{1/d}+\mu_d(B)^{1/d}$ for convex bodies $A,\,B\subseteq \R^d$, where $\mu_d$ denotes the $d$-dimensional Lebesgue measure. It is well-known that equality holds if and…

We study matrix inequalities involving partial traces for positive semidefinite block matrices. First of all, we present a new method to prove a celebrated result of Choi [Linear Algebra Appl. 516 (2017)]. Our method also allows us to prove…

Geometric and functional Brunn-Minkowski type inequalities for the lattice point enumerator $\mathrm{G}_n(\cdot)$ are provided. In particular, we show that $$\mathrm{G}_n((1-\lambda)K + \lambda L + (-1,1)^n)^{1/n}\geq…

In comparing geodesics induced by different metrics, Audenaert formulated the following determinantal inequality $$\det(A^2+|BA|)\le \det(A^2+AB),$$ where $A, B$ are $n\times n$ positive semidefinite matrices. We complement his result by…

We revisit and comment on the Harnack type determinantal inequality for contractive matrices obtained by Tung in the nineteen sixtieth and give an extension of the inequality involving multiple positive semidefinite matrices.

For positive semidefinite matrices $A$ and $B$, Ando and Zhan proved the inequalities $||| f(A)+f(B) ||| \ge ||| f(A+B) |||$ and $||| g(A)+g(B) ||| \le ||| g(A+B) |||$, for any unitarily invariant norm, and for any non-negative operator…

We show that for any log-concave measure $\mu$ on $\mathbb{R}^n$, any pair of symmetric convex sets $K$ and $L$, and any $\lambda\in [0,1],$ $$\mu((1-\lambda) K+\lambda L)^{c_n}\geq (1-\lambda) \mu(K)^{c_n}+\lambda\mu(L)^{c_n},$$ where…

Let $A,\;B$ be the positive semidefinite matrices. A matrix version of the famous Powers-St{\o}rmer's inequality $$2Tr(A^\alpha B^{1-\alpha})\geq Tr(A+B-|A-B|),\;\;\;0\leq\alpha\leq 1,$$ was proven by Audenaert et. al. We establish a…