The constraints are not convex. Consider the example below in which x1 and x2 are each vectors of 3 elements which satisfy the inequality in question, as shown. 0.5(x1 + x2) does not satisfy the inequality, thereby proving it is not convex.
>> x1 = [0.868417827606570 0.121582145814843 0.010000025679806];
>> x2 = [0.017508300926335 0.264007818119070 0.718483881445100];
>> x3 = 0.5*(x1 + x2)
0.442963064266453 0.192794981966956 0.364241953562453
>> log(x1(2)) - 0.5*(log(x1(1)) + log(x1(3)))
>> log(x2(2)) - 0.5*(log(x2(1)) + log(x2(3)))
>> log(x3(2)) - 0.5*(log(x3(1)) + log(x3(3)))
In order to use a convex programming method, you could do something like Difference of Convex Functions Programming. That would entail doing a first order (linear) expansion of the right-hand-side terms in the non-convex constraint about their current (or an initial) value. You can only use first order expansion because the quadratic term would go in the wrong direction relative to having an overall convex constraint. You will need to start the iteration with initial values (expansion points). Then update the expansion point on each subsequent iteration with the optimal value of the convex program just solved. There is no guarantee of convergence to anything, let alone to a local or global optimum.
Alternatively, you could throw it in a global optimizer, such as BARON, perhaps using a slightly positive lower bound of p so as to avoid difficulties at 0.