Let $V(x)$ be a non-decreasing, convex function on $\mathbb{R}^+$. Fix some positive integers $L$ and $k \leq L$. Let $A_{L,k}$ be the space whose elements are sequences of positive integers $r= (r_i)_{i=1}^{N}$ such that $L - \sum\limits^N_{i=1} r_i < k$ and all the $r_i \geq k$.
Prove (or disprove) that the minimum of $$ \mathcal{H}(r) := \sum\limits_{i=1}^{N} V(r_i) $$ among the sequences $r \in A_{L,k}$ is attained by the sequences which have all $r_i =k$.
If we interpret $V(x)$ as the ''cost'' required for performing a jump of length $n$, the previous expression tells us that convexity of the cost function implies that it is always more convenient to perform "small" than "large" jumps to cover a given distance.