Shortest Distance between three vertices of cones

Greetings CVX Community:
My Optimization problem is as follow
I know that problem is CONVEX, I am not able to formulate the problem in CVX.
Can anybody help?

How do you know the constraint is convex?

It is a canonical region ( Cones ).

r and theta are constants

Please provide your proof that it is convex, not a vague statement about cones.

Each constraint of the three constraints is a norm cones ( Second order Cones) which is by default Convex, do you need a mathematical proof for its convexity ?

Please show me the formulation as a second order cone. If it really is a second order cone, it should be easy to enter into CVX.

If it were just z^2 on the RHS with z >= 0, this could be entered as norm([x-xn y-yn]) <= z .If the values of r and theta are such that you are willing to impose
(z-r*cot(theta))*tan(theta) >= 0, then you are all set (or always <= 0, in which case reverse the sign of the LHS); otherwise, not.