Convex hull
Named by 14 essays across 5 fields — each of them below, with the objects they name alongside it.
One dimension up, and the circles disappear
The Delaunay triangulation is defined by a condition about circles, which is awkward to compute and awkward to reason about. Lift every point onto a paraboloid and the circles turn into planes, the condition turns into convexity, and a two-dimensional problem is solved by looking at a three-dimensional shape from underneath.
When the sites are not the same size
Give every site a weight and the boundaries slide. The cells stay convex and the edges stay straight, which is surprising, and one thing happens that the unweighted diagram never allows — a site can end up owning nothing at all.
The shape described from outside
A convex shape can be given by its boundary or by the family of lines that touch it, and the second description turns the constant-width condition into one line of arithmetic — after which the perimeter falls out, and curves with no corners at all can simply be written down.
The solid whose corners are triangulations
Take the triangulations of a hexagon as points and join two of them when a single diagonal can be swapped for another. The result is not merely a graph — it is the edge skeleton of a genuine convex polyhedron, with fourteen corners, three square faces and six pentagonal ones.
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.
Sixteen polygons with one dot inside
Fix one of Pick's two counts at one and ask what is left. The answer is a finite list, the list has exactly sixteen entries, each one is its own kind of object with a dual that is another entry, and the whole classification is a search a page can carry out.
Averaging down the triangle
Change one word in the rule that builds Pascal's triangle — take a share of each entry above instead of adding them — and the triangle stops counting and starts averaging. The same rule then draws smooth curves from polygons and approximates every continuous function by polynomials, at a rate that no amount of smoothness can improve.
Nearly the most means nearly round
A shape that holds almost as much as a circle of the same perimeter must almost be a circle. Bonnesen made that exact: the ring between a convex shape's largest inscribed circle and smallest enclosing circle is never wider than √(L² − 4πA)/π. Three quite different shapes holding 99% of the circle's area all have rings under 9.55 wide, and not one of 200 random convex shapes breaks the bound.
The roots of the slope stay inside
Mark the roots of a polynomial in the complex plane and stretch a band around them. However the roots are arranged, the roots of the derivative land inside the band — never outside, never on a new frontier. The reason is a balance of pushes, the same reason makes the derivative of a cubic mark the foci of an ellipse nobody asked for, and a question about how far inside the roots must sit has been open since 1958.
The polygon a lattice becomes from far away
Walk the grid of whole-number points with a fixed set of moves and the places reachable in r moves fill a shape. With axis steps it is a diamond, add a diagonal and it is a hexagon, move like a knight and it is a ragged thing full of holes — which, seen from far enough away, is an octagon exactly. The generators decide the polygon, and the polygon decides the count.
Seven points split three ways
Any seven points in the plane can be divided into three groups whose convex hulls share a point, and six points in general position never can. The theorem is Helge Tverberg's; its topological version, in which straight lines may bend, is true when the number of groups is a power of a prime and false otherwise — and the proof that decides which is the same antipodal argument that halves a sandwich.
A cube through a hole in a cube
Prince Rupert of the Rhine bet that a hole could be cut through a cube large enough for an equal cube to pass through it, and won. The trick is to look at the cube along a direction near its long diagonal, where its shadow is a hexagon wide enough to hold a square face with room to spare: a cube 6% larger can pass. All five Platonic solids turn out to have such a passage, found by searching over the directions from which a solid is seen — and in 2025 a solid was found that has none.
The corners of a random hull
Drop ten thousand points at random into a triangle and stretch a band round them: about twenty points touch it. Drop them into a disc and about seventy-three do. The count grows like a logarithm in one shape and like a cube root in the other, and the reason is where the boundary bends.
Lines that leave half on each side
Scatter ten points on a page and draw every line through two of them that leaves four on each side. There are at least five such halving lines and at most thirteen, and every point lies on an odd number of them. Both facts come from turning a line through half a revolution, the argument that halves two shapes with one cut. How many halving lines n points can have is the k-set problem, and its answer lies somewhere between a little more than n and n to the four-thirds.
Named alongside it
The objects these essays reach for when they reach for this one.
ConvexityDualityExhaustive searchCircleEllipseFourier seriesLatticeLifting mapPolynomialVoronoi diagramApproximationAssociativity