Homeomorphisms mapping half spaces to closed convex cones

Question

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ be a homeomorphism. I'm looking for conditions on $f$ that are sufficient to ensure that for any probability mass function, $p:\mathbb{R}^n\rightarrow[0,1]$, $D_p=\left\{f(g)\vert p.g\geq0\right\}\subseteq \mathbb{R}^n$ is a closed convex cone that contains the positive orthant and excludes the negative orthant, i.e.,

1. If $g \geq 0$ then $g\in D_p$, where $g=\left<g_1,...,g_n\right>\geq 0$ if and only if $g_i\geq0$ for all $i\leq n$.
2. If $g < 0$ then $g\not\in D_p$, where $g < 0$ if and only if $g_i<0$ for all $i\leq n$.
3. If $g\in D_p$ and $\lambda>0$ then $\lambda g \in D_p$
4. If $f, g\in D_p$ then $f + g\in D_p$
5. If $g+\epsilon\in D_p$ for all $\epsilon>0$ then $g\in D_p$

As an example of such a homeomorphism, let $f_i:\mathbb{R}^n\rightarrow\mathbb{R}$ be defined by $f_i(g)$=[$g_i/\alpha_i$ if $g_i<0$, $g_i/\beta_i$ if $g_i\geq0$] with $\alpha_i>\beta_i>0$. Let $f(g)=\left<f_1(g),...,f_n(g)\right>$. It is straightforward to verify that $D_p$ satisfies 1-5 for any probability mass function $p$. But is there a known class of homeomorphisms that map half spaces to convex cones in this way?