Question: Is $\inf\_{i \in I} \|x + i \| = \inf\_{i,j \in I} \|x + i + j \| $ and $\inf\_{i,j \in I} \| x + i + j \| = \inf\_{j \in I} \inf\_{i \in I} \| x + i + j \|$? I tried with a few different sets of real numbers for $I$ and it seems to hold. And what about $\|ab + ai + bj + ij\|$? Does it hold that $\inf\_{i,j \in I}\|ab + ai + bj + ij\| = \inf\_{i' \in I} \|ab + i'\|$? Here $x,a,b$ are some fixed real numbers. I assume if all the equalities here hold for real numbers then they also hold for complex numbers. 

In other words I am asking if the following hold:

A) $\inf\_{i \in I} \|x + i \| = \inf\_{i,j \in I} \|x + i + j \| $

B) $\inf\_{i,j \in I} \| x + i + j \| = \inf\_{j \in I} \inf\_{i \in I} \| x + i + j \|$

What if, in a more general setting, now $I$ is an ideal in some ring. Then

C) $\inf\_{i,j \in I}\|ab + ai + bj + ij\| = \inf\_{i' \in I} \|ab + i'\|$?

Answer: 
Both $\inf\_{i \in I} \|x + i \| = \inf\_{i,j \in I} \|x + i + j \| $ and $\inf\_{i,j \in I} \| x + i + j \| = \inf\_{j \in I} \inf\_{i \in I} \| x + i + j \|$ are correct provided that $0\in I$ and $i+j\in I$, whenever $i,j\in I$.

Note that, if $I\_1\subset I\_2$, then $$
\inf\_{i \in I\_1} \|x + i \|\ge\inf\_{i \in I\_2} \|x + i \|.
$$

So if $0\in I$ and $i+j\in I$, whenever $i,j\in I$, then
$$
\{i+j: i,j\in I\}=I,
$$
and hence $\inf\_{i \in I} \|x + i \| = \inf\_{i,j \in I} \|x + i + j \| $.
At least in the real number case $I=\mathbb{R}$, use triangle inequality

$$\inf\_{i,j \in I = \mathbb{R}} \|x + i + j \|\le \inf\_{i \in \mathbb{R}} \|x +i\| +\inf\_{j \in \mathbb{R}}\|j \| = \inf\_{i \in \mathbb{R}} \|x +i\| $$

The strict equality follows because the norm is nonnegative so at best we want it to be precisely zero, which we can do if we have zero in the set $I$. This may not be the case, in general though (suppose we are working with some odd polyhedron in $\mathbb{R}^2$, like a unit octogon centered at $(\pi,e)$). And if it doesn't hold in $\mathbb{R}^2$, why should it in $\mathbb{C}$, necessarily?