  It has been shown that deep neural networks of a large enough width are
universal approximators but they are not if the width is too small. There were
several attempts to characterize the minimum width $w_{\min}$ enabling the
universal approximation property; however, only a few of them found the exact
values. In this work, we show that the minimum width for $L^p$ approximation of
$L^p$ functions from $[0,1]^{d_x}$ to $\mathbb R^{d_y}$ is exactly
$\max\{d_x,d_y,2\}$ if an activation function is ReLU-Like (e.g., ReLU, GELU,
Softplus). Compared to the known result for ReLU networks,
$w_{\min}=\max\{d_x+1,d_y\}$ when the domain is $\smash{\mathbb R^{d_x}}$, our
result first shows that approximation on a compact domain requires smaller
width than on $\smash{\mathbb R^{d_x}}$. We next prove a lower bound on
$w_{\min}$ for uniform approximation using general activation functions
including ReLU: $w_{\min}\ge d_y+1$ if $d_x<d_y\le2d_x$. Together with our
first result, this shows a dichotomy between $L^p$ and uniform approximations
for general activation functions and input/output dimensions.
