d, if a center \(\bc\) is updated to \(\hat{\bc}\), then the new distance from \(\bx\) to \(\hat{\bc}\) is bounded from below and above by</p>
<p class="formulaDsp">
\[ \|\bx - \bc\|_p - \|bc - \hat\bc\|_p \leq \|\bx - \hat{\bc}\|_p \leq \|\bx - \hat{\bc}\|_p + \|\bc + \hat{\bc}\|_p. \]
</p>
<p>This allows to maintain an upper bound on the distance of \(\bx_i\) to its current center \(\bc_{q_i}\) and a lower bound to any other center \(\bc\):</p>
<p class="formulaDsp">
\begin{align*} UB_i &amp; \leftarrow U