Recently I have a conjecture on decomposing a linear program into smaller ones. I have tested it in Mathmatica by a lot of examples. However, I cannot prove it. I will appreciate if someone can give some ideas.



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> Let $f\_1,\dotsc,f\_r \in \mathbb{R}[x\_1,...,x\_n]$ ($r>n+1$) be linear functions. Then $\{f\_i \ge 0|\ 1 \le i \le r \}$ has a solution if and only if $\{f\_i \ge 0|\ 1 \le i \le r, i \ne j\}$ has a solution for each $j=1,\dotsc,r$.
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a linear programming problem
The statement in the gray box is an immediate consequence of [Helly's theorem](https://en.wikipedia.org/wiki/Helly%27s_theorem): for each $i$, the set $\{ v\in\mathbb{R}^n\colon\ f\_i(v)\geq 0\}$ is convex, and because of $r>n+1$, the condition given on the right-hand side of the equivalence implies that the intersection of any $n+1$ of these sets is inhabited, hence by Helly's theorem, the intersection of all the sets is inhabited too.


