On separation of points from additive subgroups of $l_{p}^{n}$ by linear functionals and positive definite functions
Volume 237 / 2017
Abstract
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.
