Before revision:  For arbitrary-length intervals, we show any c-approximation to \alpha , and thus also \beta , requires \Omega ] \left( \frac{n^{1/c}{2^c}} space.
After revision:  For arbitrary-length intervals, we show any c-approximation to \alpha , and thus also \beta , requires \Omega ], our result implies a separation between the insertion-only and turnstile model. For unit-radius disks, we obtain a \left( \frac{n^{1/c}{2^c}} space.