How to solve a single dot product inequality?

Question

We have an inequality of real numbers: $$ a_1x_1 + a_2x_2 + \dots + a_nx_n \ge 0. $$

Is there a way to express the most precise bounds $\{l_i$, $u_i\}$, in function of the $\{x_j\}$, such that $\forall i, a_i \in [l_i,u_i]$ ?

Answer 1

To expand on Erick Wongs answer in the comments, imagine that $n = 2$ or $3$ so that you can draw a picture. Let's say $n = 3$ for now. $x := (x_1, x_2, x_3)$ is a point in 3-dimensional space. The set of all vectors $(b_1, b_2, b_3)$ such that $b_1x_1 + b_2x_2 + b_3 x_3 = 0$ is an infinite plane through the origin: it is the plane of vectors that are orthogonal to the line passing through $0$ and $(x_1, x_2, x_3)$. Let's call this plane $P$. Now the set of points $(a_1, a_2, a_3)$ you describe make up HALF OF ALL SPACE, namely all points that are on the same side of $P$ as point $x$ is. Now look only at the first coordinates that are possible for points in this half space. For almost all choices of $x$ (that is: of $P$) they can be any number. Only in the special case that $x$ is of the form $(x_1, 0, 0)$ we have that the half space consists of all points $(a_1, a_2, a_3)$ where $a_1 \geq 0$ with no further conditions on $a_2$ and $a_3$ (or, if $x_1 < 0$, where $a_1 \leq 0$ with no conditions on $a_2, a_3$).

So the answer is one of two cases:

A. $(-\infty, \infty)$ for all coordinates

B. $(-\infty, \infty)$ for all but one coordinates and for one special coordinate either $(-\infty, 0]$ or $[0, \infty)$.