Identification of two hypercubes in $\mathbb{R}^d$

Question

Use the notation $k = (k_1,\ldots, k_d) \in \mathbb{R}^d$. How to see that the regions $$ \Big \{ \, \, k \in \mathbb{R}^d \, \, : \sum_{i=1}^d \cos(k_i) > 0 \, \, \Big \} \cap \Big \{ k \in \mathbb{R}^d : \| k \|_\infty < \pi \Big \} $$ and $$ \Big \{ k \in \mathbb{R}^d : \| k \|_1< \pi \Big \} $$ are the same (if true)?

Answer 1

This seems not to be true for $d\ge 3$. In the case $d=3$, your second set is an octahedron, while your first set is the intersection of a cube with this region:

This cannot give an octahedron.