Two algorythms constructing optimal convex approximations are grounded. The one which is iterative interprets an optimal object as providing a minimum for the middle quadratic deviation from given function values in given nodes under condition of convexity. Another algorythm is finite. It interprets an optimal approximation as theclosest to one providing unconditional minimum for the middle quadratic deviation. On this way the simplified necessary and sufficient criteria for positive definite and semidefinite matrices are obtained as auxilliary results.