Let $X$ be a finite dimensional Banach space and let $Y\subset X$ be a hyperplane. Let $\text{L}{}_Y=\{L\in \text{L}(X,Y):L{\mid }_Y=0\}$. In this note, we present sufficient and necessary conditions on $L_0\in \text{L}{}_Y$ being a strongly unique best approximation for given $L\in \text{L}\left(X\right)$. Next we apply this characterization to the case of $X=l_{\infty }^n$ and to generalization of Theorem I.1.3 from [12] (see also [13]).

We investigate in this paper the properties of some dilatations or contractions of a sequence (αn)n≥1 of Lr-optimal quantizers of an ${ℝ}^d$-valued random vector $X\in L^r\left(\mathbb{P}\right)$ defined in the probability space $(\Omega ,𝒜,\mathbb{P})$ with distribution ${\mathbb{P}}_X=P$. To be precise, we investigate the Ls-quantization rate of sequences ${\alpha }_n^{\theta ,\mu }=\mu +\theta ({\alpha }_n-\mu )=\{\mu +\theta (a-\mu ),a\in {\alpha }_n\}$ when $\theta \in {ℝ}_{+}^{☆},\mu \in ℝ,s\in (0,r)$ or s ∈ (r, +∞) and $X\in L^s\left(\mathbb{P}\right)$. We show that for a wide family of distributions, one may always find parameters (θ,µ) such that (αnθ,µ)n≥1 is Ls-rate-optimal. For the Gaussian and the exponential distributions we show...