  We have investigated analytically and numerically the liquid-glass transition
of hard spheres for dimensions $d\to \infty $ in the framework of mode-coupling
theory. The numerical results for the critical collective and self
nonergodicity parameters $f_{c}(k;d) $ and $f_{c}^{(s)}(k;d) $ exhibit
non-Gaussian $k$ -dependence even up to $d=800$. $f_{c}^{(s)}(k;d) $ and
$f_{c}(k;d) $ differ for $k\sim d^{1/2}$, but become identical on a scale
$k\sim d$, which is proven analytically. The critical packing fraction
$\phi_{c}(d) \sim d^{2}2^{-d}$ is above the corresponding Kauzmann packing
fraction $\phi_{K}(d)$ derived by a small cage expansion. Its quadratic
pre-exponential factor is different from the linear one found earlier. The
numerical values for the exponent parameter and therefore the critical
exponents $a$ and $b$ depend on $d$, even for the largest values of $d$.
