Can someone suggest the way to convexify the problem 1/x^2, where x is the decision variable (Mod note: nonconvexity is for non-scalar version of this)

Nonconvex
1 
cvx_begin

variable X(4);
loss =1./(( pi*X.*X))
minimize(loss))

cvx_end

2

Your expression does not evaluate to a scalar. But that could easily be fixed.

More seriously 1/(x'*x) is neither convex nor concave when length(x) >= 2. For instance, the Hessian of 1/(x(1)^2+x(2)^2) has one negative and one positive eigenvalue at x(1)=x(2)=1.

Any convexification of this would be a completely different model, which is not the purview of forum readers to formulate.

Note 1/x^2 is convex for scalar x, in which case pow_p(x,-2) can be used.