Suppose $A\in S_{++}^n$, so how to prove the convexity of of summation of abosolute values of this matrix? Do we need to take the second derivative ? Namely, $$\|A\|_1^{\text{off}} = \sum_{i\ne j}|a_{ij}|.$$
So what is $$\frac{\partial^2 \|A\|_1^{\text{off}}}{\partial A^2}$$