I start with some known preliminaries on the problem:

**Classical result.** The one-dimensional Cauchy functional equation
$$
\forall x,y \in \mathbb{R}, \,\,\,f(x+y)=f(x)+f(y)
$$
with $f:\mathbb{R}\to \mathbb{R}$ is only solved by the trivial solutions $f(x)=cx$, for some $c \in \mathbb{R}$, if $f$ satisfies for some additional conditions, e.g., continuity.

**Classical result with restricted domain.** Now let $\mathbb{R}^+:=(0,\infty)$. It is clear from the proof of the above classical result that if $f:\mathbb{R}^+\to \mathbb{R}^+$ is a continuous function such that
$$
\forall x,y \in \mathbb{R}^+, \,\,\,f(x+y)=f(x)+f(y) \, ,
$$
then there exists $c \in \mathbb{R}^+$ such that $f(x)=cx$ for all $x$.

**Multidimensional Cauchy functional equation.** It is also well known that if $f:\mathbb{R}^2\to \mathbb{R}$ is a continuous function such that
$$
\forall x,y \in \mathbb{R}^2, \,\,\,f(x+y)=f(x)+f(y),
$$
then there exist $A,B \in \mathbb{R}$ such that $f(x,y)=Ax+By$ for all $(x,y) \in \mathbb{R}^2$.

I know that the following generalization holds true as well. In particular, I already know how to prove it, by using a variant of the classical proof. In the following, a cone $C\subseteq \mathbb{R}^2$ is a set for which $\alpha x+\beta y \in C$ whenever $\alpha,\beta \in \mathbb{R}^+$ and $x,y \in C$.

Fact.Let $C\subseteq \mathbb{R}^2$ be a non-empty cone and $f:C \to \mathbb{R}$ be a continuous function such that $$ \forall x,y \in C, \,\,\,f(x+y)=f(x)+f(y). $$ Then there exist $A,B \in \mathbb{R}$ such that $f(x,y)=Ax+By$ for all $(x,y) \in C$.

Is it a known result? In such case, does anyone have a reference for this result?

An Introduction to the Theory of Functional Equations and Inequalities), and that much is also known for the restricted Cauchy equation in $\mathbf R^n$ (Section 13.6 in Kuczma's book). $\endgroup$ – Salvo Tringali Mar 7 '17 at 9:18complexvector spaces the "multidimensional Cauchy equation" is also a special case of Thm 7.2 in Risteski and Covachev'sComplex vector functional equations(2002). For more about the solution to the Cauchy matrix functional equation they refer to Gheorghiu,C. R. Acad. Sci. Paris256, 3562 (1963) and Kuwagaki,J Math Soc Japan14, 359 (1962). $\endgroup$ – Jules Lamers Mar 7 '17 at 10:12therestricted Cauchy equation"). $\endgroup$ – Salvo Tringali Mar 7 '17 at 17:21