How can I prove convexity of a constraint set in MATLAB?

Question

How can I prove convexity of this set in MATLAB or PYTHON?

$$\{(y_1,y_2)\in\Bbb R^2\,\mid\,\forall (x_1,x_2)\in\Bbb R^2, 2x_1^4+x_2^4+y_1x_1x_2^3+y_2x_1^3x_2>0\}$$

Is there any pre-defined library in MATLAB or PYTHON which we can define a set in it?

Answer 1

Here is a simple "done by hand" proof. For fixed $(x_1,x_2) \in \mathbb R^2$, the set $$ C_{x_1,x_2} := \{(y_1, y_2) \in \mathbb R^2 \mid 2x_1^4 + x_2^4 + x_1x_2^3y_1 + x_1^3x_2 y_2 > 0\} $$ is convex, since it is just the solution set of a linear inequality. The set in question is $$ \bigcap_{(x_1,x_2) \in \mathbb R^2} C_{x_1,x_2}. $$ Thus, it is convex since it can be written as an intersection of convex sets.