The general linear equation on open connected sets

Fix non-zero reals α₁, … , α_n with n≥ 2 and let K be a non-empty open connected set in a topological vector space such that ∑ _i≤nα_iK⊆ K (which holds, in particular, if K is an open convex cone and α₁, … , α_n> 0). Let also Y be a vector space over F: = Q(α₁, … , α_n). We show, among others, that a function f: K→ Y satisfies the general linear equation ∀x1,…,xn∈K,f(∑i≤nαixi)=∑i≤nαif(xi)if and only if there exist a unique F-linear AX→ Y and unique b∈ Y such that f(x) = A(x) + b for all x∈ K, with b= 0 if ∑ _i≤nα_i≠ 1. The main tool of the proof is a general version of a result Radó and Baker on the existence and uniqueness of extension of the solution on the classical Pexider equation.