deletions | additions       diff --git a/paragraph_Symmetry_and_reduction_begin__.tex b/paragraph_Symmetry_and_reduction_begin__.tex  index 453166a..6315387 100644  --- a/paragraph_Symmetry_and_reduction_begin__.tex  +++ b/paragraph_Symmetry_and_reduction_begin__.tex 
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\item The hemisphere can be bijectively mapped to the equatorial plane $\mathbf{e}_n^{\bot}$ with the coordinates $\mathbf{w} \in \mathbb{B}^{n-1}$ in the unit ball, by   \[ \mathbf{q}(\mathbf{w}) = (\mathbf{w}, \sqrt{1-\|\mathbf{w}\|_2^2}). \]  \item It suffices to study $g(\mathbf{w}, \mathbf{X}_0) := f(\mathbf{q}(\mathbf{w}); \mathbf{X_0})$ over the set $\{\mathbf{w}: $\Gamma := \{\mathbf{w}:  \|\mathbf{w}\|_2 < \sqrt{\frac{4n-1}{4n}} \}$. \end{enumerate}