Under what conditions on $a,b,c$ is the sum $a^3 + b^3 + c^3$ strictly greater than $(a+b+c)^2$?

Question

I was solving a problem earlier and if this was true in general it would've made everything much easier, but it's not. So I thought it would be interesting to know when exactly it is true.

Answer 1

Note that $a^3 + b^3 + c^3$ is homogeneous of order $3$ while $(a+b+c)^2$ is homogeneous of order $2$. So for any $a,b,c>0$, the statement will be true for $(ta,tb,tc)$ if $t$ is sufficiently large and false if $t>0$ is sufficiently small. The boundary between the two regions is a certain surface. Here is a picture of it.