On separation of points from additive subgroups of $l_{p}^{n}$ by linear functionals and positive definite functions

Wojciech Banaszczyk¹ ; Robert Stegliński²

¹ Faculty of Mathematics and Computer Science University of Łódź 90-238 Łódź, Poland
² Institute of Mathematics Faculty of Technical Physics Information Technology and Applied Mathematics Łódź University of Technology 90-924 Łódź, Poland

Studia Mathematica, Tome 237 (2017) no. 1, pp. 57-69

Résumé

Let $X$ be a finite-dimensional real normed space, and $K$ a closed additive subgroup of $X$. Let $a\in X\setminus K$ and let $d_X(a,K)$ be the distance from $a$ to $K$. We say that a linear functional $f\in X^*$ separates $a$ from $K$ if $d_\mathbb {R}(f(a),f(K)) \gt 0$. We say that a continuous positive definite function $\varphi :X\to \mathbb {C}$ separates $a$ from $K$ if $\varphi $ is constant on $K$ and $\varphi (a)\not =\varphi (0)$. We consider the following question: how well can $a$ be separated from $K$ by linear functionals and positive definite functions? We introduce certain quantities, denoted by $\mathit {wd}_{X}(a,K)$ and $\mathit {pd}_{X}(a,K)$, which measure the ‘distance’ from $a$ to $K$ with respect to linear functionals and positive definite functions, respectively. Then we define \[ \operatorname {wp}(X) := \sup\frac {\mathit {pd}_{X}(a,K)} {\mathit {wd}_{X}(a,K)}, \ \hskip 1em \operatorname {ps}(X) := \sup\frac {d_X(a,K)} {\mathit {pd}_{X}(a,K)}, \] the suprema taken over all closed subgroups $K\subset X$ and all $a\in X\setminus K$. We give some estimates of $\operatorname {wp}(X)$ and $\operatorname {ps}(X)$, mainly for $X=l_p^n$. In particular we prove that $\operatorname {wp}(l_p^n) \asymp _n n^{\max \{1/2,1/p\}}$ if $1\le p\le \infty $, and $\operatorname {ps}(l_p^n) \asymp _n n^{1/2}$ if $2\le p \lt \infty $. The results may be treated as finite-dimensional analogs of those obtained in Banaszczyk and Stegliński (2008, Sec. 5) for diagonal operators in $l_p$ spaces.

Détail

DOI : 10.4064/sm8559-10-2016

Keywords: finite dimensional real normed space nbsp closed additive subgroup setminus k distance nbsp say linear functional * separates mathbb say continuous positive definite function varphi mathbb separates varphi constant varphi varphi consider following question separated linear functionals positive definite functions introduce certain quantities denoted mathit mathit which measure distance respect linear functionals positive definite functions respectively define operatorname sup frac mathit mathit hskip operatorname sup frac k mathit suprema taken closed subgroups subset setminus estimates operatorname operatorname mainly n particular prove operatorname asymp max infty operatorname asymp infty results may treated finite dimensional analogs those obtained banaszczyk stegli ski sec nbsp diagonal operators spaces