\[mainTheorem\] Let $\nu$ be a mixture model with density $\rho$ of the form which we assume satisfies Assumptions \[Assump1\]. For $\sigma\in(0,\pi/4)$, suppose $${\mathcal{S}}<\frac {w_{min}(1-\cos^2(\sigma))}{w_{max} \cos^2(\sigma) N^2  }\,.$$ Define $$\tau:=4  \left( 
\sqrt{\frac{{\Theta}(1 -N{\mathcal{S}})}{{\mathcal{C}}}  } - \frac{\sqrt{ N {\mathcal{S}}}}{(1- {\mathcal{S}})} \right)^{-1} +  \sqrt{{\mathcal{S}}}\,.$$ and $$\delta^*:= \frac{w_{max} \cos^2(\sigma) N^2 {\mathcal{S}}}{w_{min}(1-\cos^2(\sigma))}\,.$$ Here, $w_{max}:=\max_{i  =1, \dots, N} w_i$ and $w_{min}:=\min_{i  =1, \dots, N} w_i$.

Suppose that $$\label{Ndelta<1}
 \tau-\sqrt{S}>0, \quad \tau N<1,$$ and $s, t >0$ satisfy $$\label{muconehyp}
\frac{t^2\sin^2(s)}{N w_{max}} \geq N\left(\frac{\tau-\sqrt{{\mathcal{S}}}}{2}\right)^2
+ 4N^{3/2} \left(\frac{1}{\sqrt{1-N \tau}} -1 \right)\,,\qquad
s+\sigma <\frac{\pi}{4}.$$ Then, the probability measure $\mu:=F_\sharp\nu$ has an orthogonal cone structure with parameters $\left(\sigma+s,\delta+t^2, \frac{1-\sin(s)}{\sqrt{w_{max}}}\right)$ for any $\delta\in[\delta^*,1)$. Here, $F$ is the continuum Laplacian embedding defined in .

Notice firstly that the smaller the quantities ${\mathcal{S}}$ and ${\mathcal{C}}/ {\Theta}$ are (i.e. the better separated the mixture model is), the smaller $\tau$ is. As a consequence the right-hand side of is also small. This allows a choice of parameters $s,t$ close to zero. Secondly, for small ${\mathcal{S}}$, we can choose $\sigma$ to be small, and if ${\mathcal{S}}$ is small compared to $\frac {w_{min}(1-\cos^2(\sigma))}{w_{max} \cos^2(\sigma) N^2  }$, then $\delta^*$ will be small also. Following Definition \[def:contcones\], small choices of $\delta$ and $t$ mean that a bigger proportion of the 