 Using a new geometric analysis, we establish lower bounds of \Omega( \epsilon ^{-d/2}) for the lightness and \Omega( \epsilon ^{-(d-1)/2}) for the sparsity of such spanners in Euclidean d-space for all d\geq 2.
<meaning-changed>  Using a new geometric analysis, we establish lower bounds of \Omega( \epsilon ^{-d/2}) for the lightness and \Omega( \epsilon ^{-(d-1)/2}) for the sparsity of such spanners in Euclidean d-space for all d\geq 2.
 Using a new geometric analysis, we establish lower bounds of \Omega( \varepsilon ^{-d/2}) for the lightness and \Omega( \epsilon ^{-(d-1)/2}) for the sparsity of such spanners in Euclidean d-space for all d\geq 2.