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Let $a$ be a vector in $\mathbb R^n$, and let $c$ be a real number. Is there a simple characterization of the set $\{x\in\mathbb R^n : (a,x) \geq c\}$ where $(a,x)$ is the inner product $\sum_{i=1}^n a_i x_i$.

Well, I'm sure that characterizations exist - but what are they? I am looking in particular for a "computer-friendly" representation. (I have a bunch of such $a$ and $c$, and I want ultimately to intersect sets of the form above, or to find a set [with a simple representation] containing the intersection but not "too much" else.)

Thank you!

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These sets are exactly half-spaces; the set $\{x\in\mathbb{R}^n:(a,x)=c\}$ (note that the inner product is usually just written as the dot product $a\cdot x$) is a hyperplane in $\mathbb{R}^n$, and your set consists of all the points to one side of this hyperplane. The intersection of a number of such sets is a convex body, but unfortunately there isn't necessarily a very efficient way of getting a description of this body for large values of $n$. The magic words that you want for finding more information about systems of equations of the form you've written are Linear Programming, and in particular you may want to have a look at the Simplex Method for one approach to dealing with the general problem.