The drive to the train station is four miles. At 35 mph it takes 6 minutes 51 seconds. At 45 it takes 5 minutes 20 seconds. That is 91 seconds saved for a 29% increase in speed and a 65% increase in the kinetic energy you carry into whatever you hit. Go from 45 to 55 and you save another 57 seconds. From 55 to 65, another 40. The same 10 mph buys less every time, because time is distance over speed, and that curve flattens.

Curvature is the whole story, and it has two parts. How the curve bends, and whether the output is something you want more of or less of. Options trading has taught everyone that convex means good, which is only true in one of the four cases.

	Output is a benefit	Output is a cost
Curve bends up	Accelerating returns. Small bounded inputs, unbounded upside.	Accelerating harm. Where ruin lives.
Curve bends down	Diminishing returns. The first unit does most of the work.	Diminishing harm. Once it is gone it cannot be taken again.

Accelerating returns.

Cold outreach, publishing, asking. Fixed small cost per attempt, no ceiling on any single hit.
Early-stage investing, where the portfolio works only because of the shape.

Accelerating harm.

Falling. Damage rises with the square of height, so a thousand falls from one foot cost a thousand units and one fall from a thousand feet costs a million. Same total distance, three orders of magnitude apart.
Queues. Average wait at 80% utilization is four in line, at 90% it is nine, at 95% nineteen, at 99% ninety-nine. A calendar booked to 95% is not 5% worse than one booked to 90%.
Missed connections. Missing the flight by two minutes and missing it by twenty cost the same six hours.
Leverage, and any repeated bet with a ruin state. The sequence ends on one bad draw.

Diminishing returns.

Speed and time saved, per the drive above.
Sleep from four hours to six, against seven hours to nine.
Income and reported wellbeing, roughly logarithmic.

Diminishing harm.

Rain, once you are already soaked.
Reputation, after the first collapse. It was built additively over years and destroyed by one event, and the second event costs almost nothing.

Which brings up the reason any of this matters day to day. Jensen's inequality says that when a function bends, the function of the average is not the average of the function. Bend up and the average outcome is worse than the outcome at the average input. Fifty percent up followed by fifty percent down averages zero and leaves you at 75. A river four feet deep on average drowns people. An average commute of thirty minutes tells you nothing about how often you miss the train, because the cost of the tail is where all the damage sits.

The average is the wrong summary statistic whenever the curve bends, and the curve almost always bends. Most of the numbers people plan around are averages.