Generator

A chord of x², and the curve under it

A generator in the analysis library, called 37 times across 7 essays. Below: what it draws with nothing chosen 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.

With nothing chosen

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.

A line under every point of x²

One minimum, or several

4 points on x², and their average

The two means, and the logarithm that separates them

Two convex sets 1.50 apart, and the line that separates them

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.

A has three to eight corners ×1
A's corners are all on its own hull, so the set is convex ×1
and a function that is not convex fails the same test, so the test has content ×1
and all but a handful of borderline cases are one or the other ×1
and every corner of the second strictly on the other ×1
and never more than three ×1
and positive against the target, which is the refutation ×1
and the other has more than one ×1
and the second is outside it ×1
and with every three assumed, the whole family has a common point ×1
B has three to eight corners ×1
B's corners are all on its own hull, so the set is convex ×1
between ninety and thirty-six hundred directions swept ×1
between one and six targets ×1
between six and twenty points in the scatter ×1
between twenty and four thousand random quadruples ×1
between two and eight values are averaged ×1
between two and seven points are averaged ×1
between two and six touch points, all inside the interval ×1
both ends of the chord are inside the interval drawn ×1
every corner of the first set is strictly on one side ×1
every point is inside the interval drawn ×1
every point of the hull lies in the hull of three of the points ×1
every three of the family have a common point ×1
every two of the family have a common point ×1
every value is positive and inside the interval drawn ×1
no line in any of the 720 directions separates the two sets ×1
no vector is both a non-negative combination and refuted by a direction ×1
one positive weight per point ×1
so a refuting direction exists for the second ×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 conjugate of x² is p²/4, found by search ×1
the convex function has exactly one local minimum on the interval ×1
the crescent's hull swallows its own mouth, so the crescent is not convex ×1
the curve stays above the tangent everywhere, not only nearby ×1
the direction is non-positive against every generator ×1
the first target is inside the cone ×1
the function drawn is a convex one ×1
the function is one this family draws ×1
the gap the line leaves is exactly the distance between the sets ×1
the geometric mean is at most the arithmetic mean ×1
the great majority of quadruples were non-degenerate ×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 scatter has a two-dimensional hull ×1
the search examined some pair of ends ×1
the search runs over between 20 and 200 candidate ends ×1
the small set lies outside the crescent ×1
the transform is drawn for a convex function ×1
the two generators are independent ×1
the two means agree exactly when every value is the same ×1
the two parts' hulls share the point the dependence produces ×1
the two sets are disjoint ×1
the view is one the family draws ×1
the weakened hypothesis is on or off ×1
the weights add to one ×1
transforming twice returns the original function ×1
two functions, side by side ×1
two generating vectors ×1
while the sweep does find a wall when one exists, so it is not simply blind ×1
with only the pairs assumed, the whole family can have nothing in common ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Analysis

A line under every point

The chord above the curve is one definition of convexity. There is a second — a line under the curve at every point, staying under everywhere — and it is the one that turns a statement about a derivative at a point into a statement about the whole function.

Analysis

A wall between two bodies

Two convex sets that do not meet can be told apart by a single straight line, and the line is a certificate — one object, checkable in a moment, proving something about every point of both. Remove convexity from either and no line exists, which is what the hypothesis was for.

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.

Analysis

The function seen from its tangents

A convex function is the upper envelope of its own tangent lines, so it can be described by giving, for each slope, how far the line of that slope has to be pushed down. That description is a second function, and applying the construction twice returns the original.

Algebra

The square that cannot be negative

Cauchy–Schwarz is the load-bearing inequality of the whole subject and is nearly always stated without proof. It is one line away from a fact nobody would argue with, and the line is a parabola with no room to cross the axis.

Analysis

Three points, however many there are

A point inside the hull of a thousand points is inside the hull of three of them. Any four points split into two groups whose hulls meet. And a family of convex sets, every three of which have a common point, has one common to all — three, in each case, being one more than the dimension.

Analysis

Where the guarantee stops

Convexity converts every downhill method into a correct one, and its absence removes the guarantee entirely rather than degrading it. What is left is a collection of partial answers, and knowing which of them apply to a given problem is most of what non-convex optimisation is.

The whole library · What the figures prove