The stresses on centrally symmetric complexes and the lower bound theorems

In 1987, Stanley conjectured that if a centrally symmetric Cohen–Macaulay simplicial complex Δ of dimension d-1 satisfies h i (Δ)=d for some i≥1, then j all j≥i. Much more recently, Klee, Nevo, Novik, and Zheng polytope P d g (∂P)=d i-d i-1 d/2≥i≥1, j-d j-1 d/2≥j≥i. This note uses stress spaces to prove both these conjectures.