 Solution

The scalar logistic loss $u\mapsto -u+\log (1+e^u)$ is strictly convex because its second derivative is $e^u/(1+e^u{)}^2>0$. If $\hat{\beta }$ and $\tilde{\beta }$ are minimizers but $X\hat{\beta }\ne X\tilde{\beta }$, strict convexity of the loss as a function of the fitted vector and convexity of the L1 norm make the objective at their midpoint strictly smaller than the common minimum. This contradiction proves thatfor every pair of solutions.