Generator

convex

A generator in the analysis library, called 7 times across 1 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

convex is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

A chord of x², and the curve under it. The curve x² with one chord drawn across it, the region between them shaded, and the midpoint heights of both marked. The comparison is computed at four hundred sample points.

`show: "jensen"`

4 points on x², and their average. Points marked on the curve x², their weighted centroid drawn as a single point, and the point of the curve directly below or above it, with both heights computed.

`show: "amgm"`

The two means, and the logarithm that separates them. The concave logarithm with a list of values marked on it, the mean of the logarithms and the logarithm of the mean drawn at different heights, and both means marked on the axis.

`show: "fail"`

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

between two and eight values are averaged ×1
between two and seven points are averaged ×1
both ends of the chord are inside the interval drawn ×1
every point is inside the interval drawn ×1
every value is positive and inside the interval drawn ×1
one positive weight per point ×1
some chord of this function dips below it ×1
the average of the values is at least the value at the average ×1
the average of the values is at most the value at the average ×1
the chord runs left to right ×1
the chord stays above the curve at every sampled point ×1
the chord stays below the curve at every sampled point ×1
the function is one this family draws ×1
the geometric mean is at most the arithmetic mean ×1
the mean of the logarithms is at most the logarithm of the mean ×1
the mean of the logarithms names the geometric mean ×1
the search examined some pair of ends ×1
the search runs over between 20 and 200 candidate ends ×1
the two means agree exactly when every value is the same ×1
the view is one the family draws ×1
the weights add to one ×1

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Analysis

The curve of the average, and the average of the curve

A curve that bends upwards keeps every one of its chords above it. That single fact, applied to a weighted average instead of a midpoint, turns into an inequality that produces the arithmetic–geometric mean inequality, Cauchy–Schwarz and the entropy bound as special cases.

The whole library · What the figures prove