Given the probability measure $\nu$ over the given region $\Omega \subset {ℝ}^n$, we consider the optimal location of a set $\Sigma$ composed by $n$ points in $\Omega$ in order to minimize the average distance $\Sigma \mapsto {\int }_{\Omega }\mathrm{dist}(x,\Sigma )\mathrm{d}\nu$ (the classical optimal facility location problem). The paper compares two strategies to find optimal configurations: the long-term one which consists in placing all $n$ points at once in an optimal position, and the short-term one which consists in placing the points one by one adding at each step at most one point and preserving...

Given the probability measure ν over the given region $\Omega \subset {ℝ}^n$, we consider the optimal location of a set Σ composed by n points in Ω in order to minimize the average distance $\Sigma \mapsto {\int }_{\Omega }\mathrm{dist}(x,\Sigma )\mathrm{d}\nu$ (the classical optimal facility location problem). The paper compares two strategies to find optimal configurations: the long-term one which consists in placing all n points at once in an optimal position, and the short-term one which consists in placing the points one by one adding at each step at most one point and preserving...

Let $A={\left(a_{n,k}\right)}_{n,k\ge 1}$ be a non-negative matrix. Denote by $L_{v,p,q,F}\left(A\right)$ the supremum of those $L$ that satisfy the inequality $${\parallel Ax\parallel }_{v,q,F}\ge L{\parallel x\parallel }_{v,p,F},$$ where $x\ge 0$ and $x\in l_p(v,F)$ and also $v={\left(v_n\right)}_{n=1}^{\infty }$ is an increasing, non-negative sequence of real numbers. If $p=q$, we use $L_{v,p,F}\left(A\right)$ instead of $L_{v,p,p,F}\left(A\right)$. In this paper we obtain a Hardy type formula for $L_{v,p,q,F}\left(H_{\mu }\right)$, where $H_{\mu }$ is a Hausdorff matrix and $0<q\le p\le 1$. Another purpose of this paper is to establish a lower bound for $\parallel A_W^{NM}{\parallel }_{v,p,F}$, where $A_W^{NM}$ is the Nörlund matrix associated with the sequence $W={\left\{w_n\right\}}_{n=1}^{\infty }$ and $1<p<\infty$. Our results generalize some works of Bennett, Jameson and present authors....

A new lower bound for the Jung constant $JC\left(l^{\left(\Phi \right)}\right)$ of the Orlicz sequence space $l^{\left(\Phi \right)}$ defined by an N-function Φ is found. It is proved that if $l^{\left(\Phi \right)}$ is reflexive and the function tΦ’(t)/Φ(t) is increasing on $(0,{\Phi }^{-1}\left(1\right)]$, then $JC\left(l^{\left(\Phi \right)}\right)\ge \left({\Phi }^{-1}(1/2)\right)/\left({\Phi }^{-1}\left(1\right)\right)$. Examples in Section 3 show that the above estimate is better than in Zhang’s paper (2003) in some cases and that the results given in Yan’s paper (2004) are not accurate.

In this paper we consider some matrix operators on block weighted sequence spaces $l_p(w,F)$. The problem is to find the lower bound of some matrix operators such as Hausdorff and Hilbert matrices on $l_p(w,F)$. This study is an extension of papers by G. Bennett, G.J.O. Jameson and R. Lashkaripour.

For 1 < p < 2 we obtain sharp lower bounds for the uniform norm of products of homogeneous polynomials on $L_p\left(\mu \right)$, whenever the number of factors is no greater than the dimension of these Banach spaces (a condition readily satisfied in infinite-dimensional settings). The result also holds for the Schatten classes ${}_p$. For p > 2 we present some estimates on the constants involved.

We prove trace inequalities of type $\left|\right|u^{\text{'}}{\left|\right|}_{L^2}^2+{\sum }_{j\in \mathbb{Z}}{k}_j|u\left(a_j\right){|}^2{\ge \lambda \left|\right|u\left|\right|}_{L^2}^2$ where $u\in H^1\left(ℝ\right)$, under suitable hypotheses on the sequences ${a_j}_{j\in \mathbb{Z}}$ and ${k_j}_{j\in \mathbb{Z}}$, with the first sequence increasing and the second bounded.