Tall Barn - Convexity

Let $g(k)$ = minimum total time using $k$ cows. Claim: $g$ is convex, meaning each additional cow helps less than the previous one. Intuition: with few cows, adding one makes a big difference. One cow per floor means total time equals $\sum a_i$. Adding a second cow to the busiest floor cuts that floor's time in half. Huge improvement.

With many cows, they're already well-distributed. Adding one more barely changes anything. This diminishing returns property, $g(k-1) - g(k) \geq g(k) - g(k+1)$, is exactly convexity (each additional unit helps less than the previous). The Aliens Trick applies.