  The Loewy length of the descent algebra of type D_{2m+1} is shown to be m+2,
for m \geq 2, by providing an upper bound that agrees with the lower bound in
\cite{BonnafePfeiffer2006}. The bound is obtained by showing that the length of
the longest path in the quiver of the descent algebra of D_{2m+1} is at most
m+1. To achieve this bound, the geometric approach to the descent algebra is
used, in which the descent algebra of a finite Coxeter group is identified with
an algebra associated to the reflection arrangement of the group.
