In this talk, we prove that finite CAT(0) cube complexes can be 
reconstructed from their boundary distances (computed in their 1-skeleta). This result 
was conjectured by Haslegrave, Scott, Tamitegama, and Tan (2023). The reconstruction of 
a finite cell complex from the boundary distances is the discrete version of the 
boundary rigidity problem, which is a classical problem from Riemannian geometry. In the proofs, we use the bijection between CAT(0) cube complexes and median graphs and the corner peelings of median graphs.
Joint work with J. Chalopin.