We consider the space $E = \mathcal{M}_{k,d}(\mathbb{R})$ equipped with the inner product defined by $$\forall (A, B) \in E^{2}, \quad \langle A \mid B \rangle = \operatorname{tr}\left(A^{\top} \cdot B\right)$$ We fix a vector $(u_{1}, \ldots, u_{d})$ in $\mathbb{R}^{d}$ with $\|u\| = 1$, and define $g(M) = \|M \cdot u\|$. 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 $m$ be a median of $g(X)$. Deduce that $\mathbb{E}\left((g(X) - m)^{2}\right) \leqslant 32$.
We consider the space $E = \mathcal{M}_{k,d}(\mathbb{R})$ equipped with the inner product defined by
$$\forall (A, B) \in E^{2}, \quad \langle A \mid B \rangle = \operatorname{tr}\left(A^{\top} \cdot B\right)$$
We fix a vector $(u_{1}, \ldots, u_{d})$ in $\mathbb{R}^{d}$ with $\|u\| = 1$, and define $g(M) = \|M \cdot u\|$. 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 $m$ be a median of $g(X)$. Deduce that $\mathbb{E}\left((g(X) - m)^{2}\right) \leqslant 32$.