I am doing some convex cone optimization and wonder whether the following set $K_1$ is convex or not.
Assume the following matrices are all in $\mathbb{R}^{n\times n}$ and symmetric. Let the set of all diagonally dominant matrices with non-negative diagonal to be $K_0$, i.e. $$K_0 = \{ A: A= [a_{ij}], \forall i,a_{ii} \geq \sum_{j\not = i}|a_{ij}|\}.$$
I wonder $$K =\{ S: \exists \;\text{positive diagonal} \;D, D^TSD \in K_0 \},$$
is convex or not(positive diagonal, $D_{ii}>0,\forall i, D_{ij}=0,\forall i\not= j)$
So far, what I have tried is brutal computation and this question seems to be that whether $$\exists d_i, d_s (\alpha a_{ss}+(1-\alpha) b_{ss}) \geq \sum_{j\not=s } ( d_s (\alpha a_{sj}+(1-\alpha) b_{sj} ) $$
for any matrices $A=[a_{ij}],B=[b_{ij}] \in K$ and $\alpha \in [0,1]$. I have no clue in choosing the $d_i$s.
Any comments or ideas?