Central-cut ellipsoid volume bound ($\mathrm{vol}(E^{\prime })/\mathrm{vol}(E)<e^{-1/(2(n+1))}$)

For $n\ge 2$, the standard ellipsoid covering a centrally cut half of the unit ball has center $e_1/(n+1)$ and shape matrix $D^{\prime }=\frac{n^2}{n^2-1}(I-\frac{2}{n+1}{e}_1{e}_1^T)$. Its eigenvalues are $n^2/(n+1{)}^2$ along $e_1$ and $n^2/(n^2-1)$ in the other directions. Hence its volume ratio is $R=\frac{n}{n+1}(\frac{n^2}{n^2-1}{)}^{(n-1)/2}$. Using $\log (1-a)<-a$ and $\log (1+b)<b$ gives $\log R<-1/(2(n+1))$. Affine changes of coordinates preserve the volume ratio. The resulting geometric convergence underlies the ellipsoid method; exact rational feasibility additionally needs input-dependent size, precision and lower-dimensional feasibility bounds.