Suppose that we have $n$ inequalities of this type $$ a_1\leq x_1+y\leq b_1\\ a_2\leq x_2+y\leq b_2\\ \vdots\\ a_n\leq x_n+y\leq b_n\\ $$ where $a_1,...,a_n$ and $b_1,...,b_n$ are finite real numbers; $x_1,..., x_n, y$ are unknowns.

Let $\Theta\equiv \{(x_1,...,x_n,y)\in \mathbb{R}^{n+1}: \text{ the inequalities above are satisfied}\}$.

Claim: $\Theta$ is not bounded.

Is this claim correct and, if yes, how can we show it?


1 Answer 1


If there is an admissible point $(x_1,x_2,\cdots x_n,y)$ then all points $(x_1+z,x_2+z,\cdots x_n+z,y-z)$ are admissible.

(The admissible set is an oblique [hyper]prism with a [hyper]rectangular base.)


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