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Let      x 1 , …  , x n x_1,\dotsc ,x_n       be linearly independent positive functions in      C ( Ω  ) C(\Omega )      , let      X X       be the vector subspace generated by the      x i x_i       and let      β  \beta       denote the curve of      R n \mathbb R^n       determined by the function      β  ( t ) = 1 z ( t ) ( x 1 ( t ) , x 2 ( t ) , …  , x n ( t ) ) \beta (t)=\frac {1}{z(t)} (x_1(t),x_2(t),\dotsc ,x_n(t))      , where      z ( t ) = x 1 ( t ) + x 2 ( t ) + ⋯  + x n ( t ) z(t)=x_1(t)+x_2(t)+\dotsb +x_n(t)      . We establish that      X X       is a vector lattice under the induced ordering from      C ( Ω  ) C(\Omega )       if and only if there exists a convex polygon of      R n \mathbb R^n       with      n n       vertices containing the curve      β  \beta       and having its vertices in the closure of the range of      β  \beta      . We also present an algorithm which determines whether or not      X X       is a vector lattice and in case      X X       is a vector lattice it constructs a positive basis of      X X      . The results are also shown to be valid for general normed vector lattices.

Finite-dimensional lattice-subspaces of 𝐶(Ω) and curves of ℝⁿ