@epsilontik, I assume that you mean that just 6 positive reals are given (corresponding to the distances between certain 4 points of the Euclidean plane.
The answer is NO--such data cannot determine the convexity because the same 6 positive reals can stand for two completely different 4-point sets, where in one case none of the points belongs to the convex hall of the remaning three, while in the other case one will. After connecting these points, in each case separately, by straight intervals, the (non-intersecting) diagonals in the non-convex case will have lengths equal to the lengths of two of the sides in the other case.
I thought I see an example by continuously deforming one configuration into another, with an intermediate one giving a solution. Now I don't see it anymore, and even have serious doubts about it, sorry. (Please, remove the undeserved by me vote for my premature answer :-)). Sorry. (I'll patiently use paper and pen next time, for verification, before rushing my happy announcement).