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I do not know if there is a name for this problem. However, we can view this in terms of the usual operator norm.

Given $A \in \mathrm{Mat}_{k \times n}(\mathbb{R}), B \in \mathrm{Mat}_{k \times m}(\mathbb{R})$, consider for $x \in \mathbb{R}^n, y \in \mathbb{R}^m$:

\begin{align*} \underset{\substack{ \lvert \lvert x \rvert \rvert = 1 \\ \lvert \lvert y \rvert \rvert = 1}}{\mathrm{sup}} \lvert \lvert Ax + By \rvert \rvert &= \underset{\substack{ \lvert \lvert x \rvert \rvert \leq 1 \\ \lvert \lvert y \rvert \rvert \leq 1}}{\mathrm{sup}} \lvert \lvert Ax + By \rvert \rvert \\ &= \underset{\substack{ \lvert \lvert x \rvert \rvert \leq 1 \\ \lvert \lvert y \rvert \rvert \leq 1}}{\mathrm{sup}} \, \left \lvert \left \lvert \begin{pmatrix}A & B\end{pmatrix} \begin{pmatrix}x \\ y\end{pmatrix} \right \rvert \right\rvert \\ &= \underset{\left \lvert \left \lvert \begin{pmatrix} x \\ y \end{pmatrix} \right \rvert \right \rvert_\infty = 1}{\mathrm{sup}} \left \lvert \left \lvert \begin{pmatrix}A & B\end{pmatrix} \begin{pmatrix}x \\ y\end{pmatrix} \right \rvert \right\rvert, \end{align*} where $\left \lvert \left \lvert \begin{pmatrix} x \\ y \end{pmatrix} \right \rvert \right \rvert_\infty := \max(\lvert \lvert x \rvert \rvert, \lvert \lvert y \rvert \rvert)$ is the supremum norm from viewing $\mathbb{R}^{n+m}$ as the product $\mathbb{R}^n \times \mathbb{R}^m$. Observe then that the last expression is $\lvert \lvert (\begin{smallmatrix}A & B\end{smallmatrix}) \rvert \rvert_{\mathrm{op}}$, where $\lvert \lvert \cdot \rvert \rvert_{\mathrm{op}}$ is the operator norm induced from $\lvert \lvert \cdot \rvert \rvert_\infty$ on $\mathbb{R}^{n+m}$ and $\lvert \lvert \cdot \rvert \rvert$ on $\mathbb{R}^k$.

I do not know if there is a name for this problem. However, we can view this in terms of the usual operator norm.

Given $A \in \mathrm{Mat}_{k \times n}(\mathbb{R}), B \in \mathrm{Mat}_{k \times m}(\mathbb{R})$, consider for $x \in \mathbb{R}^n, y \in \mathbb{R}^m$:

\begin{align*} \underset{\substack{ \lvert \lvert x \rvert \rvert = 1 \\ \lvert \lvert y \rvert \rvert = 1}}{\mathrm{sup}} \lvert \lvert Ax + By \rvert \rvert &= \underset{\substack{ \lvert \lvert x \rvert \rvert \leq 1 \\ \lvert \lvert y \rvert \rvert \leq 1}}{\mathrm{sup}} \lvert \lvert Ax + By \rvert \rvert \\ &= \underset{\substack{ \lvert \lvert x \rvert \rvert \leq 1 \\ \lvert \lvert y \rvert \rvert \leq 1}}{\mathrm{sup}} \, \left \lvert \left \lvert \begin{pmatrix}A & B\end{pmatrix} \begin{pmatrix}x \\ y\end{pmatrix} \right \rvert \right\rvert \\ &= \underset{\left \lvert \left \lvert \begin{pmatrix} x \\ y \end{pmatrix} \right \rvert \right \rvert_\infty = 1}{\mathrm{sup}} \left \lvert \left \lvert \begin{pmatrix}A & B\end{pmatrix} \begin{pmatrix}x \\ y\end{pmatrix} \right \rvert \right\rvert, \end{align*} where $\left \lvert \left \lvert \begin{pmatrix} x \\ y \end{pmatrix} \right \rvert \right \rvert_\infty := \max(\lvert \lvert x \rvert \rvert, \lvert \lvert y \rvert \rvert)$ is the supremum norm from viewing $\mathbb{R}^{n+m}$ as the product $\mathbb{R}^n \times \mathbb{R}^m$. Observe then that the last expression is $\lvert \lvert (\begin{smallmatrix}A & B\end{smallmatrix}) \rvert \rvert_{\mathrm{op}}$, where $\lvert \lvert \cdot \rvert \rvert_{\mathrm{op}}$ is the operator norm induced from $\lvert \lvert \cdot \rvert \rvert_\infty$ on $\mathbb{R}^{n+m}$ and $\lvert \lvert \cdot \rvert \rvert$ on $\mathbb{R}^k$.

Also note that this characterization $\underset{\lvert \lvert x_i \rvert \rvert = 1}{\mathrm{sup}} \, \lvert \lvert \sum_i A_i x_i \rvert \rvert = \lvert \lvert (\begin{smallmatrix} A_1 & \ldots & A_\ell \end{smallmatrix}) \rvert \rvert_{\mathrm{op}}$ works for any finite sum.

I do not know if there is a name for this problem. However, we can view this in terms of the usual operator norm.

Given $A \in \mathrm{Mat}_{n \times k}(\mathbb{R}), B \in \mathrm{Mat}_{m \times k}(\mathbb{R})$, consider for $x \in \mathbb{R}^n, y \in \mathbb{R}^m$:

\begin{align*} \underset{\substack{ \lvert \lvert x \rvert \rvert = 1 \\ \lvert \lvert y \rvert \rvert = 1}}{\mathrm{sup}} \lvert \lvert Ax + By \rvert \rvert &= \underset{\substack{ \lvert \lvert x \rvert \rvert \leq 1 \\ \lvert \lvert y \rvert \rvert \leq 1}}{\mathrm{sup}} \lvert \lvert Ax + By \rvert \rvert \\ &= \underset{\substack{ \lvert \lvert x \rvert \rvert \leq 1 \\ \lvert \lvert y \rvert \rvert \leq 1}}{\mathrm{sup}} \, \left \lvert \left \lvert \begin{pmatrix}A & B\end{pmatrix} \begin{pmatrix}x \\ y\end{pmatrix} \right \rvert \right\rvert \\ &= \underset{\left \lvert \left \lvert \begin{pmatrix} x \\ y \end{pmatrix} \right \rvert \right \rvert_\infty = 1}{\mathrm{sup}} \left \lvert \left \lvert \begin{pmatrix}A & B\end{pmatrix} \begin{pmatrix}x \\ y\end{pmatrix} \right \rvert \right\rvert, \end{align*} where $\left \lvert \left \lvert \begin{pmatrix} x \\ y \end{pmatrix} \right \rvert \right \rvert_\infty := \max(\lvert \lvert x \rvert \rvert, \lvert \lvert y \rvert \rvert)$ is the supremum norm from viewing $\mathbb{R}^{n+m}$ as the product $\mathbb{R}^n \times \mathbb{R}^m$. Observe then that the last expression is $\lvert \lvert (\begin{smallmatrix}A & B\end{smallmatrix}) \rvert \rvert_{\mathrm{op}}$, where $\lvert \lvert \cdot \rvert \rvert_{\mathrm{op}}$ is the operator norm induced from $\lvert \lvert \cdot \rvert \rvert_\infty$ on $\mathbb{R}^{n+m}$ and $\lvert \lvert \cdot \rvert \rvert$ on $\mathbb{R}^k$.

I do not know if there is a name for this problem. However, we can view this in terms of the usual operator norm.

Given $A \in \mathrm{Mat}_{k \times n}(\mathbb{R}), B \in \mathrm{Mat}_{k \times m}(\mathbb{R})$, consider for $x \in \mathbb{R}^n, y \in \mathbb{R}^m$:

\begin{align*} \underset{\substack{ \lvert \lvert x \rvert \rvert = 1 \\ \lvert \lvert y \rvert \rvert = 1}}{\mathrm{sup}} \lvert \lvert Ax + By \rvert \rvert &= \underset{\substack{ \lvert \lvert x \rvert \rvert \leq 1 \\ \lvert \lvert y \rvert \rvert \leq 1}}{\mathrm{sup}} \lvert \lvert Ax + By \rvert \rvert \\ &= \underset{\substack{ \lvert \lvert x \rvert \rvert \leq 1 \\ \lvert \lvert y \rvert \rvert \leq 1}}{\mathrm{sup}} \, \left \lvert \left \lvert \begin{pmatrix}A & B\end{pmatrix} \begin{pmatrix}x \\ y\end{pmatrix} \right \rvert \right\rvert \\ &= \underset{\left \lvert \left \lvert \begin{pmatrix} x \\ y \end{pmatrix} \right \rvert \right \rvert_\infty = 1}{\mathrm{sup}} \left \lvert \left \lvert \begin{pmatrix}A & B\end{pmatrix} \begin{pmatrix}x \\ y\end{pmatrix} \right \rvert \right\rvert, \end{align*} where $\left \lvert \left \lvert \begin{pmatrix} x \\ y \end{pmatrix} \right \rvert \right \rvert_\infty := \max(\lvert \lvert x \rvert \rvert, \lvert \lvert y \rvert \rvert)$ is the supremum norm from viewing $\mathbb{R}^{n+m}$ as the product $\mathbb{R}^n \times \mathbb{R}^m$. Observe then that the last expression is $\lvert \lvert (\begin{smallmatrix}A & B\end{smallmatrix}) \rvert \rvert_{\mathrm{op}}$, where $\lvert \lvert \cdot \rvert \rvert_{\mathrm{op}}$ is the operator norm induced from $\lvert \lvert \cdot \rvert \rvert_\infty$ on $\mathbb{R}^{n+m}$ and $\lvert \lvert \cdot \rvert \rvert$ on $\mathbb{R}^k$.

I do not know if there is a name for this problem. However, we can view this in terms of the usual operator norm.

Given $A \in \mathrm{Mat}_{n \times k}(\mathbb{R}), B \in \mathrm{Mat}_{m \times k}(\mathbb{R})$, consider for $x \in \mathbb{R}^n, y \in \mathbb{R}^m$:

\begin{align*} \underset{\substack{ \lvert \lvert x \rvert \rvert = 1 \\ \lvert \lvert y \rvert \rvert = 1}}{\mathrm{sup}} \lvert \lvert Ax + By \rvert \rvert &= \underset{\substack{ \lvert \lvert x \rvert \rvert \leq 1 \\ \lvert \lvert y \rvert \rvert \leq 1}}{\mathrm{sup}} \lvert \lvert Ax + By \rvert \rvert \\ &= \underset{\substack{ \lvert \lvert x \rvert \rvert \leq 1 \\ \lvert \lvert y \rvert \rvert \leq 1}}{\mathrm{sup}} \, \left \lvert \left \lvert \begin{pmatrix}A & B \\ 0 & 0\end{pmatrix} \begin{pmatrix}x \\ y\end{pmatrix} \right \rvert \right\rvert \\ &= \underset{\left \lvert \left \lvert \begin{pmatrix} x \\ y \end{pmatrix} \right \rvert \right \rvert_m = 1}{\mathrm{sup}} \left \lvert \left \lvert \begin{pmatrix}A & B \\ 0 & 0\end{pmatrix} \begin{pmatrix}x \\ y\end{pmatrix} \right \rvert \right\rvert, \end{align*} where $\left \lvert \left \lvert \begin{pmatrix} x \\ y \end{pmatrix} \right \rvert \right \rvert_m := \max(\lvert \lvert x \rvert \rvert, \lvert \lvert y \rvert \rvert)$ is the supremum norm from viewing $\mathbb{R}^{n+m}$ as the product $\mathbb{R}^n \times \mathbb{R}^m$. Observe then that the right-most expression is the operator norm induced from $\lvert \lvert \cdot \rvert \rvert_m$ evaluated on $\left(\begin{smallmatrix}A & B \\ 0 & 0\end{smallmatrix}\right)$.

I do not know if there is a name for this problem. However, we can view this in terms of the usual operator norm.

Given $A \in \mathrm{Mat}_{n \times k}(\mathbb{R}), B \in \mathrm{Mat}_{m \times k}(\mathbb{R})$, consider for $x \in \mathbb{R}^n, y \in \mathbb{R}^m$:

\begin{align*} \underset{\substack{ \lvert \lvert x \rvert \rvert = 1 \\ \lvert \lvert y \rvert \rvert = 1}}{\mathrm{sup}} \lvert \lvert Ax + By \rvert \rvert &= \underset{\substack{ \lvert \lvert x \rvert \rvert \leq 1 \\ \lvert \lvert y \rvert \rvert \leq 1}}{\mathrm{sup}} \lvert \lvert Ax + By \rvert \rvert \\ &= \underset{\substack{ \lvert \lvert x \rvert \rvert \leq 1 \\ \lvert \lvert y \rvert \rvert \leq 1}}{\mathrm{sup}} \, \left \lvert \left \lvert \begin{pmatrix}A & B\end{pmatrix} \begin{pmatrix}x \\ y\end{pmatrix} \right \rvert \right\rvert \\ &= \underset{\left \lvert \left \lvert \begin{pmatrix} x \\ y \end{pmatrix} \right \rvert \right \rvert_\infty = 1}{\mathrm{sup}} \left \lvert \left \lvert \begin{pmatrix}A & B\end{pmatrix} \begin{pmatrix}x \\ y\end{pmatrix} \right \rvert \right\rvert, \end{align*} where $\left \lvert \left \lvert \begin{pmatrix} x \\ y \end{pmatrix} \right \rvert \right \rvert_\infty := \max(\lvert \lvert x \rvert \rvert, \lvert \lvert y \rvert \rvert)$ is the supremum norm from viewing $\mathbb{R}^{n+m}$ as the product $\mathbb{R}^n \times \mathbb{R}^m$. Observe then that the last expression is $\lvert \lvert (\begin{smallmatrix}A & B\end{smallmatrix}) \rvert \rvert_{\mathrm{op}}$, where $\lvert \lvert \cdot \rvert \rvert_{\mathrm{op}}$ is the operator norm induced from $\lvert \lvert \cdot \rvert \rvert_\infty$ on $\mathbb{R}^{n+m}$ and $\lvert \lvert \cdot \rvert \rvert$ on $\mathbb{R}^k$.