= Solution
The <empirical risk> is
$$
f(\beta)=\frac1n\sum_{i=1}^n\log_2(1+e^{-y_ix_i^T\beta}).
$$
For $g(z)=\log_2(1+e^{-z})$,
$$
g''(z)=\frac{e^z}{\log(2)(1+e^z)^2}\geq0,
$$
so $g$ is a <convex function>. Each signed margin $y_ix_i^T\beta$ is an <affine function> of $\beta$, and composition with an affine function preserves convexity. Equivalently,
$$
\nabla^2f(\beta)=\frac1n\sum_{i=1}^ng''(y_ix_i^T\beta)x_ix_i^T
$$
is a <positive semidefinite matrix>. Therefore $f$ is convex.
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