I have the following function, $$f(x,y) = x^3 + y^3 + x^2 + y^2 + (x+y)^2 - 2,$$ my goal is to find the domain for which $f(x,y)$ is convex. I know to formally prove convexity, I can compute the hessian, $H$, and find values of $x,y$ such that all the principle leading minors are positive (sylvester's criterion for $H$ to be positive definite).
However, I am interested in knowing whether the following logic is correct:
$x^3$ and $y^3$ are strictly convex if $x,y$ > 0. Also $x^2, y^2,$ and $(x+y)^2$ are convex on $x,y>0.$ A positive sum of convex functions is convex, so $$g(x,y) = x^3 + y^3 + x^2 + y^2 + (x+y)^2$$ is convex on $x,y >0.$ Adding a constant shouldn't effect convexity, so $f(x,y)$ is convex on $x,y > 0$. The reason I am not sure whether this is true because $g(x,y) = h(x) + j(y) + k(x,y),$ is the positive sum of a single variable convex and multiple variable convex a convex function?
Yes, because a convex function in 1 variable is also convex in $x$ and $y$ (although a strictly convex function in $x$ is not strictly convex in both variables). So all 5 pieces of your $g$ are convex in both variables when $x,y>0$.
However, outside of that subdomain, you should work with the Hessian to see the exact region of convexity.