• Suppose we are trying to minimize $x^2 + y^2$ subject to $x + y = 10$.
• We can think of $(x, y)$ as two points located symmetrically about $5$, suppose it is $x = (5 + \epsilon)$ and $y = (5 - \epsilon)$.
• See that the function $f(k) = k^2$ is such that the output becomes larger as we go to the right / increase the argument than the rate at which the output becomes smaller as we go to the left / decrease the argument.
• This is clear by computing $\partial_k f = 2k$, which means that if $k_r > k_l$ (right/left), then $\partial_{k_r} f = 2 k_r$, while $\partial_{k_l} f = 2 k_l$, so if we step to the left and the right by $\epsilon$, keeping the total the same, the sum will change by $(2 k_r - 2 k_l) \epsilon > 0$.
• Said differently, because the function is convex / $f''(x) > 0$, this means that $\partial_k|_r f > \partial_k|_l f$, and thus we can trade the loss of the total from moving to the left (a $- \partial_k|_l \epsilon$ for the gain of the total from moving to the right (a $+ \partial_k|_r \epsilon$).

• Picture:

          * dx=1.2         /|---->        - |       /  |     --   |*---/     |-dx=0.8   |  <-|     |    |    x=0.6   x=0.4

• We gain more by moving rightwards (in terms of $f(r+dx) \simeq f(r) + f'(r) dx = f(r) + 2f(r)dx$ than we lose by moving leftward (in terms of $f(l-dx) \simeq f(l) - f'(l) dx = f(l) - 2f(l) dx$. Since $f(r) > f(l)$, the total we gain is still net positive.
• Said differently again, we gain faster by moving from a point that is rightwards, than the rate at which we lose from a point that is leftwards.
• Said differently again, the elevation gain is larger towards the right, so a small motion rightwards gains us more elevation than a small motion leftwards loses elevation.

§ How does this relate to convexity?

• What is the geometric intution for this being related to "below a line"?