I need your expertise in solving the following problem:
Let $D,V \in \mathbb{R}^{n \times n}$ be a diagonal matrix and an orthogonal matrix respectively and let $D^\prime,V^\prime \in \mathbb{R}^{(n+1) \times (n+1)}$ be a diagonal matrix and an orthogonal matrix respectively.
Lets denote $G = DV^T$ and $G^\prime = D^\prime {V^\prime}^T$.
It is known that:
There exists $v \in \mathbb{R}^d$ such that for every $x \in \mathbb{R}^d$, $$ \left| \left| G(x - v) \right| \right|_2 \geq \frac{\left| \left| G^\prime \begin{bmatrix}x \\ 1 \end{bmatrix}\right| \right|_2 }{2}$$
$\left| {G^\prime}^{-1} \right| = const(d^2) \cdot \left| G \right|$
Is it possible to prove that for every $x \in \mathbb{R}^d$ where $x \neq v$, $$ \left| \left| G(x - v) \right| \right|_2 \leq const(d^2) \cdot \left| \left| G^\prime \begin{bmatrix}x \\ 1 \end{bmatrix}\right| \right|_2 ?$$
Please advice and thanks in advance.