Price equilibria with transfers

I am pretty new to Microeconomics so please bear with me if I am missing something obvious. I am solving the following problem:

The question is in the setting of pure exchange economy with two consumers A, and B who have the utility functions:

$$u_A(x_{1A}, x_{2A}) = x_{1A} x_{2A}$$, $$u_B(x_{1B}, x_{2B}) = x_{1B} + 2x_{2B}$$. It is assumed that the endowments of both customers is positive for both commodities.

(a) Sketch the Edgeworth box, indifference curves of the consumers and the Pareto set where $$\bar{\omega_1} = \omega_{1A} + \omega_{2A} = \omega_{1B} + \omega_{2B} = \bar{\omega_2}$$.

I have solved this part and the Edgeworth box looks like this:

Next it is asked (b) wether all the Pareto Optima of this economy can be sustained as price quasi-equilibria with transfers. I know that I should us the second welfare theorem but I am not sure how I can check the convexity of the preferences using utilities.

• The preferences represented by a utility function are convex if and only if the utility function is quasi-concave. Here, the utility functions are even concave. – Michael Greinecker Dec 14 '20 at 22:21