IIUC, The paper's notation S^(d-1) means the unit sphere in R^d (e.g., the familiar unit circle is S^1 living in R^2). So, i think, x in the algorithm is already a unit vector.
Reference:
Section 2:Preliminaries
...
We use the notation S^d−1 to denote the hypersphere in R^d of radius 1.
Section 3.1
Let x ∈ S^d−1 be a (worst-case) vector on the unit sphere in dimension d.
Right but in reality IIUC w ∈ R^d and it's x = w / ||w|| ∈ S^(d-1) and then given r = x - Qmse^-1( Qmse( x ) ) the scalar you use is derived as ||r|| (I'm missing a couple subscript twos there I think).
I was primarily aiming to confirm my understanding given the author's omission but also the scalar is subtly different than in your linked explanation (although conceptually equivalent).
Reference: Section 2:Preliminaries ... We use the notation S^d−1 to denote the hypersphere in R^d of radius 1.
Section 3.1 Let x ∈ S^d−1 be a (worst-case) vector on the unit sphere in dimension d.