Where this rung has been, and where it lands
This rung opened with convex bodies and the supporting hyperplanes that touch them from outside, moved to polytopes and the face arithmetic of Euler and Dehn-Sommerville, weighed volume with Brunn-Minkowski and mixed volumes, and arithmetized space with Minkowski's lattice theorem. This last guide is where the threads tie off. We take up two questions that sound like they belong in a kitchen: how tightly can you stack equal balls, and how little do you need to check to know a whole family of convex sets shares a point? The first is sphere packing; the second is the trio Helly-Radon-Carathéodory.
These two themes look like strangers, yet both turn on a single hidden constant: in R^n, convexity is ruled by the number n+1. A point of a convex hull is already an average of at most n+1 of the original points; a family of convex sets meets globally the instant every n+1 of them meet; and the only packings we can actually prove optimal live in dimensions (8 and 24) where exceptionally rigid lattices happen to exist. Keep n+1 in mind — it is the quiet protagonist of everything below.
Carathéodory: the convex hull is built from small committees
Begin with the most frugal of the three. Carathéodory's theorem states: if a point x lies in the convex hull of a set S in R^n, then x already lies in the convex hull of at most n+1 points chosen from S. However sprawling S is — a million scattered dots, an entire region — every point inside its hull is a weighted average of a small committee of size n+1. In the plane (n=2) that means every point of a convex hull sits inside some triangle whose three corners belong to S; in space (n=3) inside some tetrahedron.
Why n+1 and not fewer? The proof is a clean dimension count. Suppose x is written as a convex combination of m points with m > n+1. Then those m points are affinely dependent, because R^n holds at most n+1 affinely independent points — so there is a nontrivial relation sum c_i = 0 with sum c_i v_i = 0, not all c_i zero. Slide the weights along this relation, increasing or decreasing them in lockstep, until one weight first hits zero; you have dropped a point while x stays a convex combination of the rest. Repeat until only n+1 points remain. The relation you exploited is exactly Radon waiting in the wings.
Radon and Helly: when an intersection is forced
Radon's theorem is the engine room. Any n+2 points in R^n can be partitioned into two disjoint groups whose convex hulls intersect. In the plane this is a four-point fact: any 4 points fall into one of exactly two patterns — three forming a triangle with the fourth inside (split the inner point against the triangle), or four in convex position (split into the two crossing diagonals). Either way, two sub-hulls overlap. The proof reuses the affine-dependence relation from Carathéodory: with n+2 points the relation sum c_i = 0, sum c_i v_i = 0 must exist; put the points with positive coefficients in one group, the negative ones in the other, rescale so both sides sum to 1, and the common value of the two combinations is the witness point.
From Radon, Helly's theorem follows by induction on the number of sets. Helly says: among finitely many convex sets in R^n, if every n+1 of them have a common point, then all of them share a common point. The local-to-global leap is the striking part — you never inspect the whole family at once, only its (n+1)-element sub-families. On the line (n=1) it reduces to a fact you can see by hand: pairwise-overlapping intervals share a point. Take the largest left endpoint a and the smallest right endpoint b; 'every two overlap' forces a <= b, and any point in [a, b] lies in all of them.
Sphere packing: an easy question with no easy answers
Now the other pole of the rung. The sphere packing problem asks for the largest fraction of R^n that congruent, non-overlapping balls can cover — the packing density. In dimension 2 the answer is the honeycomb of pennies, density pi / sqrt(12) ~ 0.9069, proved rigorously (Thue, then Fejes Tóth). In dimension 3 the answer is the grocer's pile of oranges, the face-centered-cubic stacking with density pi / sqrt(18) ~ 0.7405 — the Kepler conjecture, posed in 1611 and not settled until Hales's computer-assisted proof in 1998, fully machine-formalized only in 2014.
Here honesty is essential. Above dimension 3 almost everything is open — except two miraculous cases. In dimension 8 the densest packing is the E_8 lattice, and in dimension 24 it is the Leech lattice; both optimalities were proved only in 2016 (Viazovska for dimension 8, then Cohn-Kumar-Miller-Radchenko-Viazovska for 24). The tool is a magic certificate: an auxiliary function whose Fourier transform vanishes at exactly the right radii, forcing an upper bound that happens to meet the lattice's density on the nose. For every other dimension n >= 4 the exact optimal density is simply unknown. This loops straight back to the previous guide: these record packings are lattice packings, so the geometry of numbers is the problem's natural home.
n=2: pi/sqrt(12) ~ 0.9069 hexagonal (proved: Thue / Fejes Toth) n=3: pi/sqrt(18) ~ 0.7405 FCC / Kepler (proved: Hales) n=8: pi^4/384 ~ 0.2537 E_8 lattice (proved: Viazovska) n=24: pi^12/12! ~ 0.00193 Leech lattice (proved: CKMRV) other n>=4: optimal density UNKNOWN
Why the two halves are one subject
The bridge between packing and combinatorial convexity is the linear-programming bound. Cohn and Elkies recast packing as an optimization over functions: choose an auxiliary function with the right sign pattern and Fourier behavior, and it certifies an upper bound on the density. That is precisely the spirit of Helly and Carathéodory — replace an impossible global search with a small, checkable certificate. Viazovska's breakthrough was constructing the exactly tight certificate in dimensions 8 and 24 out of modular forms. Combinatorial convexity and analytic packing are two faces of one idea: in convex geometry, a global truth is pinned down by a finite, low-dimensional witness.
- To apply Helly in practice: phrase your goal as 'is there a single point lying in every set?', verify each set really is convex, then check only the (n+1)-fold intersections — if all of those are nonempty, the global intersection is nonempty too.
- To get a packing upper bound (Cohn-Elkies): find a function f with f(0) > 0, with f(x) <= 0 whenever |x| >= r, and with Fourier transform everywhere nonnegative; then the packing density is at most a ratio read directly off f.
- To recognize when that bound is tight: it must match a known lattice packing exactly — this happens for E_8 (n=8) and Leech (n=24), and, as far as we can prove, essentially nowhere else.
That closes the convex-and-discrete rung. You can now read a single convex body from three angles at once: as a shape pinned by supporting hyperplanes, as a volume governed by Brunn-Minkowski, and as a combinatorial object obeying the n+1 law of Helly-Radon-Carathéodory. One body, three towers — and sphere packing is exactly where all three lean on one another.
Common traps and honest limits
A few warnings before you climb on. First, the constant in each theorem is the ambient dimension's n+1 (or n+2 for Radon) and it does not depend on how many sets or points you have — that dimension-only constant is precisely what makes these results so powerful, and also precisely why they collapse the moment a set is non-convex. Second, 'densest known' is not 'densest possible': for most dimensions we hold packings nobody has beaten, not proofs that none can be. Third, Kepler and the dimension-8/24 results are genuine theorems with complete proofs, but the 8/24 proofs rest on deep modular-form constructions — a guide can state and motivate them, it cannot reproduce them, and you should be honest with yourself about that gap.
Finally, resist reading 'density decreases with dimension' as a law. It does plummet in the handful of cases we can compute, but the true optimal densities for general high n are unknown, and even the gap between the best lower and upper bounds widens as n grows. As elsewhere on this ladder, honesty about open problems — the optimal packing for nearly every dimension above 3, the exceptions 8 and 24 aside — serves a graduate reader far better than a tidy but false sense of closure.