Characterization of finite point sets determining orthogonality

Characterize the finite sets of points S in R^d, for positive integers 1 <= k < d, such that a d x d matrix A satisfying ||Ax_1 wedge ... wedge Ax_k|| = ||x_1 wedge ... wedge x_k|| for every x_1,...,x_k in S must be orthogonal, and such that every orthogonal matrix has this property.

Background

The paper proves that an affine linear map preserving the k-dimensional volumes of all k-simplices in Rd must have an orthogonal linear part when 1 <= k < d. The authors then ask whether a finite collection of points can suffice to enforce the same conclusion from preservation of the associated wedge-product norms.

The unresolved problem is to characterize exactly which finite point sets have this determining property. Such a characterization would support applications to volume rigidity, where only finitely many simplex-volume constraints are available rather than preservation of all k-simplex volumes in the ambient space.

References

In view of applications to volume rigidity (discussed below), it would be of interest to extend Proposition \ref{prop:isometries_main} to the situation when eq:prop_isometries_main holds for all $k$-subsets of a finite set of points. To the best of our knowledge, the following problem is still open. Let $d,k$ be positive integers with $1\leq k<d$. Characterise when a finite set of points $P$ in $\mathbb{R}d$ has the following property: for any $d\times d$ matrix $A$, $|Ax_1\wedge A x_2\wedge \dots \wedge A x_k|=|x_1\wedge x_2\wedge \dots \wedge x_k|$ for all $x_1, x_2,\dots, x_k\in S$ if and only if $A$ is orthogonal.

eq:prop_isometries_main:

∥Ax1∧Ax2∧⋯∧Axk∥=∥x1∧x2∧⋯∧xk∥\|Ax_1\wedge Ax_2\wedge \dots \wedge Ax_k\|=\|x_1\wedge x_2\wedge \dots \wedge x_k\|

— Volume Rigidity of Simplicial Manifolds  (2503.01647 - Cruickshank et al., 3 Mar 2025) in Section 2, immediately after Proposition 2.2 (numbering as rendered in the source), Problem environment