McMullen's characterization conjecture for simplicial polytopes

Let $(h_0,h_1,\dots,h_d)$ be a sequence of integers, and let an M-vector mean a sequence satisfying Macaulay's conditions. The associated g-vector is

$$(h_0,h_1-h_0,\dots,h_{\lfloor d/2\rfloor}-h_{\lfloor d/2\rfloor-1}).$$

McMullen's conjecture. The sequence $(h_0,h_1,\dots,h_d)$ is the h-vector of the boundary of a simplicial $d$-polytope if and only if $h_0=1$, $h_0\le h_1\le\dots\le h_{\lfloor d/2\rfloor}$, and its g-vector is an M-vector. This is the celebrated g-theorem, which the source notes was proved by Billera and Lee and by Stanley; the conjecture is therefore solved.