Clarify the domain of bound on d given by bgmlv(v-u)>=0

main.tex

@@ -1242,21 +1242,26 @@ assert bgmlv3_d_upperbound_exp_term == (
 \bgroup
 \def\psi{\chern_1^{\beta}(v)}
 \def\phi{\chern_2^{\beta}(v)}
-\begin{dmath}
+\begin{equation*}
 	\label{eqn-bgmlv3_d_upperbound}
 	d \leq
 	\sage{bgmlv3_d_upperbound_linear_term}
 	+ \sage{bgmlv3_d_upperbound_const_term_alt}
 	+ \sage{bgmlv3_d_upperbound_exp_term_alt2}
-\end{dmath}
+	\qquad
+	\text{where }r>R
+\end{equation*}
 \egroup
 
 
 \noindent
+For $r=R$, $\Delta(v-u)\geq 0$ is always true, and for $r<R$ it gives a lower
+bound on $d$, but it is weaker than the one given by the lower bound in
+subsubsection \ref{subsect-d-bound-radiuscond}.
 Viewing the right hand side of equation \ref{eqn-bgmlv3_d_upperbound}
 as a function of $r$, the linear and constant terms almost match up with the
-ones in the previous section, up to
-$\chern_2^{\beta}(v)$.
+ones in the previous section, up to the 
+$\chern_2^{\beta}(v)$ term.
 
 
 However, when specializing to problem \ref{problem:problem-statement-2} again