I am trying to prove that the following set is connected.
$ R(a,b) = \{ W \in [0,1]^{m \times m} \, | \, \, \sum_{ij=1}^m W_{ij}=a,tr(W^3)=b,\,W_{ij}=W_{ji}\} $
where $m \in \mathbb{N}$, $m \geq 2$ and $ a,b \in [0,1]$ and $tr(\cdot)$ is the matrix trace.
My partial result is:
The imagen of the set $ R(b) = \{ W \in [0,1]^{m \times m} \, | \, \, tr(W^3)=b,\,W_{ij}=W_{ji}\} $ by the transformation $f:[0,1]^{m \times m} \rightarrow [0,1]^{m \times m}$, $f(W) = W^3$ is convex.
Since if $A,B \in f(R(b))$ then $tr(\alpha A +(1-\alpha)B) = \alpha tr(A) +(1-\alpha)tr(B) =b$.
Moreover, the application $f(W)$ is a homeomorphism when $W$ is nonsigular. The set of nonsigular matrices is dense and has two connected components. Hence the preimagen of $f$ i.e. $R(b)$ is connected.
Therefore $R(a,b)$ is the intersection of a hyperplane $R(a) = \{ W \in [0,1]^{m \times m} \, | \, \, \sum_{ij=1}^m W_{ij}=a,W_{ij}=W_{ji}\} $ and $R(b)$. However I have no clue on how to prove or disprove if $R(a,b)$ is connected.
Some of you has an idea?
Thanks in advance.