2023/06/16 by Murai, Satoshi, Novik, Isabella, Zheng, Hailun · 1 citation
#05E45 #13F55 #52B05 #52C25 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2306.09816
A conjecture of Kalai asserts that for d≥ 4, the affine type of a prime simplicial d-polytope P can be reconstructed from the space of affine 2-stresses of P. We prove this conjecture for all d≥ 5. We also prove the following generalization: for all pairs (i,d) with 2≤ i≤ \lceil \frac d 2\rceil-1, the affine type of a simplicial d-polytope P that has no missing faces of dimension ≥ d-i+1 can be reconstructed from the space of affine i-stresses of P. A consequence of our proofs is a strengthening of the Generalized Lower Bound Theorem: it was proved by Nagel that for any simplicial (d-1)-sphere Δ and 1≤ k≤ \lceil(d)/(2)\rceil-1, gk(Δ) is at least as large as the number of missing (d-k)-faces of Δ; here we show that, for 1≤ k≤ \lfloor(d)/(2)\rfloor-1, equality holds if and only if Δ is k-stacked. Finally, we show that for d≥ 4, any simplicial d-polytope P that has no missing faces of dimension ≥ d-1 is redundantly rigid, that is, for each edge e of P, there exists an affine 2-stress on P with a non-zero value on e.