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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 arelay $Z$ and the relay can communicate to the destination via an error-freedigital link of rate $R_0$, while $Y$ and $Z$ are conditionally independent andidentically distributed given $X$. We develop two new upper bounds on thecapacity of this channel that are tighter than existing bounds, including thecelebrated cut-set bound. Our approach significantly deviates from the standardinformation-theoretic approach for proving upper bounds on the capacity ofmulti-user channels. We build on the blowing-up lemma to analyze theprobabilistic geometric relations between the typical sets of the $n$-letterrandom variables associated with a reliable code for communicating over thischannel. 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 orderfor 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 binarysymmetric channels. Our tighter bounds on the capacity of the relay channelimmediately 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 broadcastchannel becomes completely noisy as $p\to 1/2$ and its capacity, and thereforethe 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.

Improving on the Cut-Set Bound via Geometric Analysis of Typical Sets