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  "body": "\n## Statement\n\nLet $P$ be a simplicial convex polytope of dimension $d \\ge 1$, that is, a polytope all of whose proper faces are simplices.\nLet $(h_0, h_1, \\dots, h_d)$ be the [[h-vector|DCK64GFPD7]] of the boundary complex of $P$.\nThen\n$$h_i = h_{d-i} \\quad \\text{for } 0 \\le i \\le d.$$\n\n## Remarks\n\nOne proof uses a shelling of the boundary complex, an order of the facets in which each new facet meets the earlier ones in a union of its own facets.\nFor each facet, record the size of its smallest new face; then $h_i$ is the number of facets for which this size is $i$.\nThe reverse of a line shelling is again a shelling, and reversing the order turns size $i$ into size $d - i$, which gives the symmetry.\n\nThe equation $h_0 = h_d$ is equivalent to Euler's relation $f_0 - f_1 + f_2 - \\dots + (-1)^{d-1} f_{d-1} = 1 - (-1)^d$ for the boundary.\nFor $d = 3$ the equations give $f_1 = 3 f_0 - 6$ and $f_2 = 2 f_0 - 4$, so the number of vertices determines the whole f-vector of a simplicial $3$-polytope.\n\nExample: the octahedron has $h = (1, 3, 3, 1)$, which is symmetric.\n\nThe hypothesis that the complex is the boundary of a polytope (more generally, a sphere) is necessary.\nThe full triangle, viewed as a simplicial complex of dimension $2$, has $f = (1, 3, 3, 1)$ and $h = (1, 0, 0, 0)$, which is not symmetric.\n\n## Explanations\n\n### L1 human\n\nThe face numbers of a polytope with triangular faces are not free: they satisfy linear equations.\nIn the h-vector form the equations are a simple mirror symmetry.\nFor example, a convex polyhedron in space made of triangles with $v$ vertices always has $3v - 6$ edges and $2v - 4$ triangles.\n\n### L2 human\n\nThe equations hold more generally for simplicial spheres and for Eulerian complexes.\nThey show that about half of the entries of the f-vector of a simplicial $d$-polytope determine the rest.\nThe remaining question, which h-vectors occur at all, is answered by the [[g-theorem|4CZMWQVC30]].\n",
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