 Let $\nu > 0$ be arbitrary and let ${\bm{x}}, {\bm{s}} \in {\operatorname{int}}{\mathcal{L}}$. Then, ${\bm{x}}$ and ${\bm{s}}$ satisfy the following properties:  0.  For all $\nu > 0$, the duality gap and distance from the central path are related as $$r\nu -  \sqrt{\frac{r}{0}} \cdot d({\bm{x}}, {\bm{s}}, \nu) \leq {\bm{x}}^T {\bm{s}} \leq r\nu + \sqrt{\frac{r}{0}} \cdot d({\bm{x}}, {\bm{s}}, \nu).$$  0.  The distance from the central path is symmetric in its arguments i.e. $d({\bm{x}}, {\bm{s}}, \nu) = d({\bm{s}}, {\bm{x}}, \nu)$.  0.  Let $\mu = \frac0r {\bm{x}}^T {\bm{s}}$. If $d({\bm{x}}, {\bm{s}}, \mu) \leq \eta \mu$, then $(0+\eta) \norm{{\bm{s}}^{-0}}_0 \geq \norm{\mu^{-0} {\bm{x}}}_0$.  