Wachspress Geometry
Wachspress Geometry is a young subject in the intersection of geometry, combinatorics and algebra. It emerged from the observation of unexpected connection between several mathematical objects defined on convex polytopes. The field is distinguished by the diversity of involved disciplines, including algebraic geometry, polyhedral combinatorics, convex geometry, valuation theory, geometric rigidity, spectral graph theory; also touching on mathematical physics, geometric modeling and statistics.
The immediate goal of Wachspress Geometry is to explain and exploit these connections and to work towards a unifying explanation for the ubiquity of “Wachspress phenomena” in polytope theory. Another way to frame Wachspress Geometry is as a particularly well-understood instances of Positive Geometry. Here the goal would be to extend tools of Wachspress Geometry to more general positive geometries.
Given a convex polytope \(P\subset\Bbb R^d\), the central objects of study are constructed on \(P\) and include:
- the canonical form \(\Omega_P\), a (geometric) valuation on convex polytopes defined via polar volumes and taking on values in the rational functions.
- adjoint polynomial \(\operatorname{adj}_P\) (or adjoint for short) is the lowest degree polynomial vanishing on the polytope’s residual arrangement. Its zero locus is known as adjoint hypersurface. The polynomial also appears as the numerator in the canonical form.
- Wachspress coordinates \(\alpha_P:P\to\Delta_n\) are the unique rational barycentric coordinates of \(P\) that have the lowest possible degree. Alternative definitions are based on cone volumes in the polar dual. The adjoint appears as the denominator polynomial.
- the Wachspress variety \(V:=\alpha_P(P)\) is the image of the Wachspress coordinates. It is an algebraic variety of dimension \(d\), smooth inside the \(n\)-vertex standard simplex \(\Delta_n\), and intersects the boundary of \(\Delta\) in a polyhedral complex isomorphic to \(\partial P\). Its ideal is the Wachspress ideal which, if \(P\) is simplicial, is related to its Stanley-Reisner ideal.
- the Izmestiev matrix \(M_P\), a special Colin de Verdière matrix of the polytopes edge graph. It has a Lorentzian signature and a kernel that encodes the polytopes geometry. Its row sums yield the Wachspress coordinates.
- the Wachspres stress \(\omega_P\) is a distinguished stress that exists for all coned polytope frameworks and can be seen as a consequence of its piecewise linear structure. It certifies the rigidity (actually prestress stability) of convex coned polytope frameworks. Its clomponents are Wachspress coordinates and entries of the Izmestiev matrix.
Other related constructions of interest are the Wachspress map (in geometric modeling) and Wachspress models (in algebraic statistics).
Wachspress Geometry is at the core of my SPP 2458 project “Wachspress Coordinates – a bridge betwen Algebra, Geometry and Combinatorics”. In Septermber 2024, Rainer Sinn and me organized a workshop on “Wachspress Geometry” in Leipzig to bring together experts on the various aspects of the field.
I plan to write here more about the subject. For now I collected literature that can serves as a starting point for the interested reader:
Literature
Wachspress coordinates
- E. L. Wachspress, “Barycentric coordinates for polytopes”
- J. Warren, “Barycentric coordinates for convex polytopes”
- J. Warren, “On the Uniqueness of Barycentric Coordinates”
- T. Ju, P. Liepa, J. Warren, “A general geometric construction of coordinates in a convex simplicial polytope”
- M. S. Floater, A, Gillette, N. Sukumar, “Gradient bounds for Wachspress coordinates on polytopes”
- M. S. Floater, “Optimality of a Gradient Bound for Polyhedral Wachspress Coordinates”
- M. S. Floater, “Wachspress and mean value coordinates”
- K. Kohn, K. Ranestad, “Projective Geometry of Wachspress Coordinates”
- M. S. Floater, J. Kosinka, “On the injectivity of Wachspress and mean value mappings between convex polygons”
Canonical forms and Positive Geometry
- N. Arkani-Hamed, Y. Bai, T. Lam, “Positive Geometries and Canonical Forms”
- T. Lam, “An invitation to positive geometries”
- C. Gaetz, “Canonical forms of polytopes from adjoints”
- Y. Gao, T. Lam, L. Xue, “Dual Mixed Volume”
- C. Eur, T. Lam, “Canonical forms of oriented matroids”
The Izmestiev matrix, Wachspress stesses and rigidity theory
- I. Izmestiev, “The Colin de Verdière number and graphs of polytopes”
- M. Winter, “Rigidity, Tensegrity and Reconstruction of Polytopes under Metric Constraints”
- R. Connelly, S. J. Gortler, L. Theran, M. Winter, “The Stress-Flex Conjecturets”
- R. Connelly, S. J. Gortler, L. Theran, M. Winter, “Energies on Coned Convex Polytopes”
Adjoint polynomials and hypersurfaces
- K. Kohn, K. Ranestad, “Projective Geometry of Wachspress Coordinates”
- K. Kohn et al, “Adjoints and Canonical Forms of Polypols” (also on mathrepo)
Wachspress varieties, ideals and models