I have some real parameters here. The $\mu_i$ - for $i=1,2,3,4,5$ - are 'convex coefficents' in that $\mu_i\geq 0$ and $\sum_{i}\mu_i=1$. The $x$ and $z$ are such that $x^2+z^2\leq 1$.
The inequalities are $$\begin{align} -1 &<\mu_1-\mu_2-\mu_3+\mu_4-z\mu_5 &<1 \\-1&<\mu_1-\mu_2-\mu_3+\mu_4+z\mu_5&<1 \\ -1&<\mu_1+\mu_2+\mu_3+\mu_4-\mu_5&<1 \\ -1&<\mu_1-\mu_4+\sqrt{(\mu_2-\mu_3)^2+\frac{\mu_5^2x^2}{2}}&<1 \\-1&<\mu_1-\mu_4-\sqrt{(\mu_2-\mu_3)^2+\frac{\mu_5^2x^2}{2}}&<1 \end{align}$$
I have put these five inequalities as well as the six in the previous paragraph (along with the equation) into Mathematica's Reduce command but it doesn't seem able to handle it. Have I any hope of solving this system?
Context:
Any solutions correspond to symmetric states ($\nu=\nu\circ S$ where $S$ is the antipode) on the Kac-Paljutkin Quantum Group (something about which is said here) whose convolution powers converge to the Haar state.
Subproblems:
- It is not hard to see that these inequalities can't hold if $\mu_5=0$ for in this case we can't have $|\sum_{i}\mu_i|<1$ because $\sum_i\mu_i=1$.
- I am also interested in what happens when $x$ and $z$ are equal to zero.
