Understanding the proof of the concavity of entropy.

In the proof above it is hard for me to comprehend the equation, especially the $\log|X|$. What is the $u$, the so-called "the uniform distribution on $|X|$ outcomes"?

Any kind of suggestions for me to understand the equation or the theorem are highly appreciated.

$$u(x)=\frac{1}{|X|}, \forall x \in X$$
• Thanks very much for your detailed mathematical speculation. But could you please teach me any thing about the form of $\log(|X|)$? What is the relation between the cross entropy and that? – Lerner Zhang Nov 5 '16 at 7:43
• I know that it is for the proof, but any thing else, considering the saying of "uniform distribution on $|X|$ outcomes" Thanks. – Lerner Zhang Nov 5 '16 at 7:44
• $\log (|X|)$ is a constant isn't it? – Siong Thye Goh Nov 5 '16 at 7:52
• Let $X = \left\{1,2,3 \right\}$, $u_X(1)=u_X(2)=u_X(3)=\frac13$ which is a constant and $\log (|X|) =\log 3$. – Siong Thye Goh Nov 5 '16 at 8:00
• $|X|$ is just the cardinality of the alphabet. In example of coin tosses where you just care about head or tail, $|X|=2$. If you are interested in the outcome of a dice regardless of whether it is biased or not, $|X|=6$. – Siong Thye Goh Nov 5 '16 at 8:08