1)     Convex economy is the general equilibrium pattern where consumers have combination on the indifference curves. It is about how prices adjust to equilibrium and that general equilibrium exist only under certain conditions. It explains the demand and supply. It is a differentiable map with fundamental rules that give a smooth production
2)     Efficient allocation is pareto optimality where class of economies are decentralized.. in general equilibrium there is efficient allocation that makes production smooth that there are no further advantageous reallocation of the factors of production. Using lagrangian equation it can be proved mathematically
3)     The conclusion drawn from general equilibrium and efficient allocation of a smooth covex economy with production clearly shows how a perfect economy can be if all the factors of production are evenly allocated and a general equilibrium can be maintained.