Consider the Johnson-Lindenstrauss lemma in the case where we can assume the $n$ input points $x\_i$ in $\mathbb{R}^d$ are actually located on the sphere 
$$S^{d-1}(r):=\{u=(u\_1,\ldots,u\_{d}): u\_1^2+\cdots+u\_d^2=r\},$$
for some $r>0.$


How does this impact the conclusion of the theorem? In particular, can the upper and lower bounds
$$
(1-\varepsilon)\sqrt{k/d}~\left| x\_i-x\_j\right|^2\leq \left| f(x\_i)-f(x\_j)\right|^2 
\leq (1+\varepsilon)\sqrt{k/d}~\left| x\_i-x\_j\right|^2,
$$ 
which hold with probability at least $1-\frac{1}{n^{c-2}},$
with $f$ an orthogonal projection on a random $k-$dimensional subspace,
where $k=c \log n/\varepsilon^2,$ possibly be strengthened in this case?



Johnson-Lindenstrauss Lemma on $S^{d-1}$
Given that the worst-case configurations require dimension projection proportional to the one obtained by linear embeddings


<http://people.seas.harvard.edu/~minilek/publications/papers/jl_tight.pdf>


it's not possible to gain by restricting the points to the sphere. Furthermore, I believe in the proof of this paper the input points (except for the origin) are all unit vectors. 


