Affine Stresses: The Partition of Unity and Kalai’s Reconstruction Conjectures

Kalai conjectured that if P is a simplicial d-polytope that has no missing faces of dimension d-1 , then the graph of P and the space of affine 2-stresses of P determine P up to affine equivalence. We propose a higher-dimensional generalization of this conjecture: if 2≤ i≤ d/2 and P is a simplicial d-polytope that has no missing faces of dimension ≥ d-i+1 , then the space of affine i-stresses of P determines the space of affine 1-stresses of P. We prove this conjecture for (1) k-stacked d-polytopes with 2≤ i≤ k≤ d/2-1 , (2) d-polytopes that have no missing faces of dimension ≥ d-2i+2 , and (3) flag PL (d-1) -spheres with generic embeddings (for all 2≤ i≤ d/2 ). We also discuss several related results and conjectures. For instance, we show that if P is a simplicial d-polytope that has no missing faces of dimension ≥ d-2i+2 , then the (i-1) -skeleton of P and the set of sign vectors of affine i-stresses of P determine the combinatorial type of P. Along the way, we establish the partition of unity of affine stresses: for any 1≤ i≤ (d-1)/2 , the space of affine i-stresses of a simplicial d-polytope as well as the space of affine i-stresses of a simplicial (d-1) -sphere (with a generic embedding) can be expressed as the sum of affine i-stress spaces of vertex stars. This is analogous to Adiprasito’s partition of unity of linear stresses for Cohen–Macaulay complexes.