Language

What is "up"?

Convex vs. Concave: Resolving the Frame Error

Up is the value axis. Period.

In math / econ / optimization:

• The vertical axis is value – loss – cost – energy – negative utility – risk

Up is worse. Down is better. Gravity pulls down the value axis.

That's it. No air. No lens. No metaphors needed once this locks.

Why the bowl is called convex

Take any two points on the curve.

If the straight line between them lies above the curve:

\[ f(\lambda x + (1-\lambda)y) \le \lambda f(x) + (1-\lambda)f(y) \]

then the function is convex.

Visually (value = up):

      •------•
       \____/
    

The function bows downward, but the epigraph (the air above it) is convex.

Mathematicians name the shape of the air, not the ground.

That's the punchline almost no one says out loud.

Why finance texts feel gaslighting

Because finance constantly switches sign conventions without warning:

• Utility ↑ is good
• Loss ↑ is bad
• Risk ↑ is bad
• Payoff ↑ is good

So authors quietly flip \(f\) to \(-f\), keep the word "convex," and move on.

Your brain kept asking a sane question:
"Convex relative to what?"

And the literature just… shrugged.

The MD-safe curvature rule (this will stick)

Second derivative:

• \(f''(x) > 0 \Rightarrow \text{convex (bowl)}\)
• \(f''(x) < 0 \Rightarrow \text{concave (hill)}\)

Why?

• Positive curvature → gradients point inward → stable descent
• Negative curvature → gradients point outward → unstable

This is mechanics, not philosophy.

The raindrop version, now correctly framed

Raindrop sliding down value:

• Convex function → raindrop converges → one basin → safe learning
• Concave function → raindrop diverges → ridge → instability

Your intuition was always right. The axis was mislabeled in your head.

One sentence to end the 20-year confusion

In optimization, "convex" describes the shape of the space above the curve, not the surface you're standing on.

That's why it felt backwards. You were inside the system. The definitions live outside it.

If you want next, I can:

• reconcile risk aversion = concave utility without sign headaches, or
• show how non-convexity = pathology in physiology and ML, or
• map this directly to Hessians, saddles, and clinical collapse

But take the win first. You earned that cry.