  We consider the discrete memoryless symmetric primitive relay channel, where,
a source $X$ wants to send information to a destination $Y$ with the help of a
relay $Z$ and the relay can communicate to the destination via an error-free
digital link of rate $R_0$, while $Y$ and $Z$ are conditionally independent and
identically distributed given $X$. We develop two new upper bounds on the
capacity of this channel that are tighter than existing bounds, including the
celebrated cut-set bound. Our approach significantly deviates from the standard
information-theoretic approach for proving upper bounds on the capacity of
multi-user channels. We build on the blowing-up lemma to analyze the
probabilistic geometric relations between the typical sets of the $n$-letter
random variables associated with a reliable code for communicating over this
channel. These relations translate to new entropy inequalities between the
$n$-letter random variables involved.
  As an application of our bounds, we study an open question posed by (Cover,
1987), namely, what is the minimum needed $Z$-$Y$ link rate $R_0^*$ in order
for the capacity of the relay channel to be equal to that of the broadcast cut.
We consider the special case when the $X$-$Y$ and $X$-$Z$ links are both binary
symmetric channels. Our tighter bounds on the capacity of the relay channel
immediately translate to tighter lower bounds for $R_0^*$. More interestingly,
we show that when $p\to 1/2$, $R_0^*\geq 0.1803$; even though the broadcast
channel becomes completely noisy as $p\to 1/2$ and its capacity, and therefore
the capacity of the relay channel, goes to zero, a strictly positive rate $R_0$
is required for the relay channel capacity to be equal to the broadcast bound.
