Minkowski's lattice-point theorem
/ min-KOF-skee /
Scatter dots in a perfectly regular grid across space, then draw a symmetric blob centered at one of the dots. Minkowski's theorem says that if the blob is big enough — bigger than a definite threshold tied to the grid's spacing — it is FORCED to swallow another grid dot besides its center. You cannot make a large centrally-symmetric region avoid all the other lattice points; sheer volume compels a hit. It is the founding theorem of the geometry of numbers, turning a counting question about integers into a statement about volume.
Precisely, let L be a lattice in R^n — the set of integer combinations of n linearly independent vectors — with fundamental domain of volume d(L) (its covolume). Let K be a convex body that is symmetric about the origin (x in K implies -x in K). Minkowski's theorem states: if vol(K) > 2^n * d(L), then K contains a nonzero lattice point. The proof is a beautiful pigeonhole (Blichfeldt's argument): shrink K by half to (1/2)K; if vol((1/2)K) > d(L), then translating (1/2)K by lattice vectors and folding back into one fundamental domain forces two pieces to overlap, and the difference of the two overlapping points is a nonzero lattice point inside K. The constant 2^n is sharp — the open cube (-1, 1)^n with the integer lattice has volume exactly 2^n and contains no nonzero lattice point.
This single theorem proves deep facts in number theory almost for free: that every positive integer is a sum of four squares (Lagrange), the existence of good rational approximations (a geometric form of Dirichlet's theorem), and bounds in algebraic number theory like the finiteness of the class number via Minkowski's bound. Minkowski's SECOND theorem refines it, bounding the product of the successive minima of K (the smallest scalings at which K first contains 1, 2, ..., n independent lattice points). An honest caveat: central symmetry and convexity are both essential — drop either and the theorem fails, since a large but lopsided or non-convex region can dodge every nonzero lattice point. And the bound is on volume strictly exceeding 2^n d(L); at exactly the threshold a nonzero point need not lie in the open body.
Take L = Z^2 (the integer grid, covolume 1) and K a disk of radius r centered at the origin, area pi*r^2 — convex and symmetric. Minkowski's threshold is 2^2 * 1 = 4, so once pi*r^2 > 4, i.e. r > 2/sqrt(pi) ~ 1.128, the disk must contain a nonzero integer point. Indeed at r slightly above 1 it already catches (1,0), (0,1), etc.; the theorem guarantees a hit no matter how the body is shaped, as long as it is symmetric, convex, and big enough.
A symmetric convex body of volume > 2^n times the covolume must contain a nonzero lattice point.
Both hypotheses are indispensable: drop central symmetry (e.g. a large triangle off-center) or convexity (a thin annulus of huge area) and the body can avoid every nonzero lattice point. Minkowski's theorem is NOT 'large area forces a lattice point' — it is 'large area plus symmetry plus convexity.'