Hi

I am meeting a problem concerning semi-definite positive matrices, and I have no clue concerning them, the classical approaches I know have not given any result, maybe people used to manipulating them could help me...

Call $\mathscr{P}$ the convex set of symmetric SDP matrices $S$ of order $N\geq 1$ such that $|S_{ij}|<1$ . Consider the mapping $$\Phi:S=(S_{ij})\mapsto (\frac{2}{\pi}\arcsin(S_{ij})).$$

I think it is a (well) known fact that $\Phi(\mathscr{P})\subset \mathscr{P}$. My question is the following: Is $\Phi(\mathscr{P})$ convex?

At first I thought that the answer was "no", but I checked that for $N=2$ the answer is "yes". I have no clue as how to test higher dimensions. Does anyone have a suggestion on a method?

Another possible formulation: Given two SDP matrices $A$ and $B$, and $a\in [0,1]$, is the matrix $C$ defined by $$C_{ij}=\sin\left(a\arcsin(A_{ij})+(1-a)\arcsin(B_{ij})\right)$$ also SDP?