Properties of $T$ so that $f(T)$ is bounded.

Question

Suppose $A\subseteq\mathbb{R}^n$. And define the function $f:A\rightarrow\mathbb{R}$ as $$f(\mathbf{a})=\mathbf{a}\cdot(\mathbf{e}_j-\mathbf{e}_i)$$ where $\mathbf{e}_k=(0~0~\cdots~0~1~0~\cdots~0)^t$, the one being in the $k^{th}$ position. What properties can we assert about $A$, so that $f(A)$ is bounded.