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The '''Hilbert basis''' of a [[convex cone]] ''C'' is a minimal set of integer [[vector (mathematics)|vector]]s in ''C'' such that every [[integer]] vector in ''C'' is a [[conical combination]] of the vectors in the Hilbert basis with integer coefficients.
== Definition ==
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:<math>C=\{ \lambda_1 a_1 + \ldots + \lambda_n a_n \mid \lambda_1,\ldots,\lambda_n \geq 0, \lambda_1,\ldots,\lambda_n \in\mathbb{R}\}\subset\mathbb{R}^d,</math>
we consider the [[monoid]] <math>C\cap L</math>. By [[Gordan's lemma]], this monoid is finitely generated, i.e., there exists a [[finite set]] of lattice points <math>\{x_1,\ldots,x_m\}\subset C\cap L</math> such that every lattice point <math>x\in C\cap L</math> is an integer conical combination of these points:
:<math> x=\lambda_1 x_1+\ldots+\lambda_m x_m, \quad\lambda_1,\ldots,\lambda_m\in\mathbb{Z}, \lambda_1,\ldots,\lambda_m\geq0.</math>
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