We consider the space $E = \mathcal{M}_{k,d}(\mathbb{R})$ equipped with the Frobenius norm $\|\cdot\|_{F}$. We fix a unit vector $u$ in $\mathbb{R}^{d}$, define $g(M) = \|M \cdot u\|$, and let $C = \{M \in \mathcal{M}_{k,d}(\mathbb{R}) \mid g(M) \leqslant r\}$. Let $X = (\varepsilon_{ij})_{1 \leqslant i \leqslant k, 1 \leqslant j \leqslant d}$ be a random variable taking values in $\mathcal{M}_{k,d}(\mathbb{R})$, whose coefficients $\varepsilon_{ij}$ are independent Rademacher random variables. Let $r$ and $t$ be two real numbers, with $t > 0$. Deduce that $$\mathbb{P}(g(X) \leqslant r) \cdot \mathbb{P}(g(X) \geqslant r + t) \leqslant \exp\left(-\frac{1}{8}t^{2}\right)$$
We consider the space $E = \mathcal{M}_{k,d}(\mathbb{R})$ equipped with the Frobenius norm $\|\cdot\|_{F}$. We fix a unit vector $u$ in $\mathbb{R}^{d}$, define $g(M) = \|M \cdot u\|$, and let $C = \{M \in \mathcal{M}_{k,d}(\mathbb{R}) \mid g(M) \leqslant r\}$. Let $X = (\varepsilon_{ij})_{1 \leqslant i \leqslant k, 1 \leqslant j \leqslant d}$ be a random variable taking values in $\mathcal{M}_{k,d}(\mathbb{R})$, whose coefficients $\varepsilon_{ij}$ are independent Rademacher random variables. Let $r$ and $t$ be two real numbers, with $t > 0$. Deduce that
$$\mathbb{P}(g(X) \leqslant r) \cdot \mathbb{P}(g(X) \geqslant r + t) \leqslant \exp\left(-\frac{1}{8}t^{2}\right)$$