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 \(2\le i\le d/2\) and P is a simplicial d-polytope that has no missing faces of dimension \(\ge d-i+1\), 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 \(2\le i\le k\le d/2-1\), (2) d-polytopes that have no missing faces of dimension \(\ge d-2i+2\), and (3) flag PL \((d-1)\)-spheres with generic embeddings (for all \(2\le i\le d/2\)). 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 \(\ge d-2i+2\), 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 \(1\le i\le (d-1)/2\), 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.