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  "body": "\n## Statement\n\nLet $n \u003e d \\ge 2$, and let $P$ be a [[convex polytope|300K6CB1KH]] of dimension $d$ with $n$ [[vertices|R7V3S9V891]].\nLet $C(n, d)$ be the [[cyclic polytope|M6YBMXKJYD]] of dimension $d$ with $n$ vertices.\nThen for every $i$ with $0 \\le i \\le d - 1$,\n$$f_i(P) \\le f_i(C(n, d)),$$\nwhere $f_i$ is the number of $i$-dimensional faces, the $i$-th entry of the [[f-vector|872BAKDQQK]].\n\n## Remarks\n\nThe theorem was proved by Peter McMullen in 1970, after Theodore Motzkin had conjectured it.\nProof idea: a small perturbation of the vertices makes $P$ simplicial without lowering any $f_i$, so one may assume $P$ simplicial.\nThen one passes to the h-vector of the boundary, uses the [[Dehn–Sommerville equations|4WKHF5BCTA]] $h_i = h_{d-i}$, and proves $h_i \\le \\binom{n - d + i - 1}{i}$ for $i \\le d/2$ with a shelling of the boundary.\nCyclic polytopes attain all these bounds, because they are neighborly.\n\nThe number of facets of $C(n, d)$ is\n$$f_{d-1}(C(n,d)) = \\frac{n}{n - m} \\binom{n - m}{m} \\text{ if } d = 2m, \\qquad 2 \\binom{n - m - 1}{m} \\text{ if } d = 2m + 1.$$\nFor $d = 3$ this gives $2n - 4$, the number of triangles of any simplicial $3$-polytope with $n$ vertices.\nFor $d = 4$ and $n = 8$ it gives $8 \\cdot 15 / 6 = 20$ facets.\n\nBy [[polar polytope|6DXF1T506B]] duality, the theorem also bounds the number of vertices of a $d$-polytope with $n$ facets.\n\n## Explanations\n\n### L1 human\n\nAmong all polytopes with a given dimension and a given number of corners, which one has the most faces?\nThe upper bound theorem says: the cyclic polytope, and it wins in every dimension of face at once.\nSo there is a single shape that is the champion for edges, for two-dimensional faces, and so on.\n\n### L2 human\n\nIn the dual form, a polytope with $n$ facets in dimension $d$ has at most about $n^{\\floor{d/2}}$ vertices.\nThis bounds the number of vertices that the simplex method or a vertex enumeration algorithm may meet, and the size of convex hulls in computational geometry.\nThe theorem was extended to simplicial spheres by Richard Stanley using commutative algebra.\nIts lower-bound partner, for simplicial polytopes, is the lower bound theorem, and both are contained in the [[g-theorem|4CZMWQVC30]].\n",
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