Summary: The \(D\)-dimensional superstring theory is known to be invariant under the duality symmetry \(b\leftrightarrow \alpha'/b\), where \(b\equiv e^\beta\sqrt{\alpha'}\) is the constant radius of the \(N\)-dimensional internal space and \(\alpha'\) is the slope parameter, the \((M+1)\)-dimensional physical space-time being flat, where \(D=M+N+1\). It is shown, starting from the \(D\)-dimensional, vacuum Einstein theory, that this symmetry can be generalized to a curved space-time, with \(\beta(t)=-\alpha(t)\) and \(t\leftrightarrow t^2_0/t\), where \(t\) is comoving time, but it will only be an isometry of the (conformally transformed) physical space if this is characterized by a single radius function \(e^{\alpha(t)}\). This would explain the homogeneity, isotropy and flatness of the Universe. The dimensionalities are uniquely determined by the dimensional reduction to be \(D=10\), \(M=3\), \(N=6\). Further aspects of the duality symmetry are discussed and the notion of a negative dimension is introduced. For the closed type I superstring, density fluctuations sufficient to account for the presence of galaxies will be present, provided that \(G\mu\lesssim 10^{-6}\), where \(G\equiv M^{-2}_P\) is the Newtonian gravitational constant, \(M_P\) is the Planck mass and \(\mu=\alpha^{\prime 2}/2\pi b^6\) is the effective mass per unit length of the string. This condition agrees with the estimate \(b\approx 23M^{-1}_P\sim 23\sqrt{\alpha'}\), obtained by Klein via quantization of the electric charge in the Kaluza-Klein theory. It is also in accord with the expectation that all the higher-dimensional superstring scales are of approximately the same order of magnitude.