# Dehn–Sommerville equations

The **Dehn–Sommerville equations** are linear relations among the numbers of faces of a simplicial polytope, or, dually, of a simple polytope.<sup>[1](https://mathworld.wolfram.com/Dehn-SommervilleEquations.html)</sup> A polytope is *simplicial* if all its faces are simplices, and *simple* if every vertex lies on exactly d edges, the smallest number geometrically possible in dimension d; the two classes are dual, so a relation for one class translates into a relation for the other.<sup>[2](https://www.ms.uky.edu/~Lee/ma714fa13/notes.pdf)</sup><sup> • </sup><sup>[3](https://cdoneill.sdsu.edu/teaching/s22-621/projects.pdf)</sup> The equations were found by Max Wilhelm Dehn (1878–1952), who proved them for dimensions 4 and 5 in 1905, and by Duncan MacLaren Young Sommerville (1879–1934), who established the general case in 1927.<sup>[4](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/13483049ECEDF0F2D760C50BAE8D6F21/S0025579300002850a.pdf/div-class-title-the-maximum-numbers-of-faces-of-a-convex-polytope-div.pdf)</sup>

| Key fact | Detail |
|---|---|
| Subject | Linear relations among the face numbers f_i of a simplicial polytope, or dually of a simple polytope<sup>[1](https://mathworld.wolfram.com/Dehn-SommervilleEquations.html)</sup> |
| h-vector form | For a simplicial d-polytope, h_i(P) = h_{d−i}(P) for 0 ≤ i ≤ ⌊(d−1)/2⌋<sup>[2](https://www.ms.uky.edu/~Lee/ma714fa13/notes.pdf)</sup> |
| Polynomial form | With h(t) = Σ f_i(t−1)^i for a simple polytope, the coefficients are symmetric: h(t) = t^d h(1/t)<sup>[1](https://mathworld.wolfram.com/Dehn-SommervilleEquations.html)</sup> |
| Completeness | They are the most general linear equations satisfied by the f-vectors of all simple (or simplicial) polytopes<sup>[5](https://www.mathnet.ru/php/getFT.phtml?jrnid=rm&option_lang=eng&paperid=320&what=fullteng)</sup> |
| Dimension count | The affine hull of the f-vectors of simple d-polytopes has dimension ⌊d/2⌋<sup>[6](https://users.fmf.uni-lj.si/lavric/brondsted.pdf)</sup> |
| History | Proved by Dehn in 1905 for d = 4, 5; extended to all dimensions by Sommerville in 1927<sup>[4](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/13483049ECEDF0F2D760C50BAE8D6F21/S0025579300002850a.pdf/div-class-title-the-maximum-numbers-of-faces-of-a-convex-polytope-div.pdf)</sup> |

## The h-vector form

For a d-dimensional simple polytope, let f_i count its i-dimensional faces, including f_d = 1, and define

\[ h(t) = \sum_{i=0}^{d} f_i (t-1)^i . \]

The Dehn–Sommerville equations state that the coefficients of this polynomial are symmetric,

\[ h(t) = t^{d} h(1/t), \]

that is, the coefficient of t^i equals the coefficient of t^{d−i}.<sup>[1](https://mathworld.wolfram.com/Dehn-SommervilleEquations.html)</sup> Equivalently, for a simplicial d-polytope P the h-vector (h_0(P), …, h_d(P)) defined by

\[ h_i = \sum_{j=0}^{i} (-1)^{i+j} \binom{d-j}{d-i} f_{j-1}(P) \]

satisfies h_i(P) = h_{d−i}(P) for i = 0, …, ⌊(d−1)/2⌋, where ⌊x⌋ is the greatest integer function.<sup>[2](https://www.ms.uky.edu/~Lee/ma714fa13/notes.pdf)</sup> The f-vector and the h-vector determine each other, so the symmetry is a statement about face counts in either encoding.<sup>[7](https://www.researchgate.net/publication/2095872_Moment-angle_complexes_and_combinatorics_of_simplicial_manifolds)</sup>

The cube illustrates the simple-polytope form: its face counts are f_0 = 8, f_1 = 12, f_2 = 6, and f_3 = 1, giving h(t) = (1+t)^3, whose coefficients are visibly symmetric.<sup>[1](https://mathworld.wolfram.com/Dehn-SommervilleEquations.html)</sup> In f-vector terms, a simple 3-polytope satisfies the two independent relations 3f_0 − 2f_1 = 0 and f_0 − f_1 + f_2 = 2; the second is Euler's formula V − E + F = 2, and the first says every vertex of a simple 3-polytope lies on 3 edges.<sup>[6](https://users.fmf.uni-lj.si/lavric/brondsted.pdf)</sup>

**Euler's relation as a special case.** In the general simple-polytope relation

\[ f_k = \sum_{j=0}^{k} (-1)^{j} \binom{d-j}{d-k} f_j , \]

the case k = d is the Euler relation Σ_{j=0}^{d} (−1)^j f_j = 1, which holds for every convex d-polytope, not just simple ones.<sup>[8](https://matthbeck.github.io/papers/ccdnew.pdf)</sup> For general convex polytopes, the Euler identity is the only linear relation satisfied by the vectors of face numbers.<sup>[5](https://www.mathnet.ru/php/getFT.phtml?jrnid=rm&option_lang=eng&paperid=320&what=fullteng)</sup>

## Completeness and dimension count

The relations are not merely satisfied by all simple polytopes; they generate everything. The Dehn–Sommerville relations are the most general linear equations satisfied by the f-vectors of all simple (or simplicial) polytopes, a result proved using h-vectors by showing that the affine hull of the h-vectors of simple n-polytopes is a plane of dimension ⌊n/2⌋, with the products of simplices Δ^k × Δ^{n−k} supplying affinely independent examples.<sup>[5](https://www.mathnet.ru/php/getFT.phtml?jrnid=rm&option_lang=eng&paperid=320&what=fullteng)</sup> Brønsted's course notes compute the same dimension, ⌊d/2⌋, for the affine hull of the f-vectors of simple d-polytopes, against dimension d−1 for arbitrary d-polytopes.<sup>[6](https://users.fmf.uni-lj.si/lavric/brondsted.pdf)</sup> In other words, of the d+1 face numbers of a simple d-polytope, only about half can be chosen freely; the Dehn–Sommerville equations determine the rest.

## Role in the g-theorem and the Upper Bound Theorem

The equations form the linear part of the complete characterization of face numbers. The g-theorem, which describes the possible f-vectors of simple (or simplicial) polytopes, was conjectured by [Peter McMullen](https://www.edgepedia.org/peter-mcmullen) in 1970 and proved around 1980, with Richard Stanley establishing necessity and Louis Billera and Carl Lee establishing sufficiency. Its first condition is the Dehn–Sommerville symmetry h_i = h_{n−i}; the remaining conditions are two groups of inequalities, linear and non-linear.<sup>[5](https://www.mathnet.ru/php/getFT.phtml?jrnid=rm&option_lang=eng&paperid=320&what=fullteng)</sup> When P is simplicial, the Dehn–Sommerville relations condense the f-vector into the g-vector, which has length ⌈(d+1)/2⌉, so only about half of the entries of the f-vector need be determined.<sup>[9](https://arxiv.org/abs/1512.04513)</sup>

The symmetry also drives the Upper Bound Theorem. For cyclic polytopes, the convex hulls of finitely many points on the moment curve, the h-entries on one side of the symmetry are explicit binomial coefficients, h_i = C(n−d−1+i, i) for 0 ≤ i ≤ ⌊d/2⌋, and the Dehn–Sommerville symmetry gives the remaining entries by h_i = h_{d−i}.<sup>[10](https://androma.org/page/Geometric%20Combinatorics)</sup> In 1970, McMullen established the upper bound theorem: among all convex hulls of n points in R^d, cyclic polytopes have the greatest number of faces in each dimension, and because of this symmetry it is enough to maximize the lower half of the h-vector in order to bound every face number.<sup>[11](https://pmc.ncbi.nlm.nih.gov/articles/PMC11415421/)</sup><sup> • </sup><sup>[10](https://androma.org/page/Geometric%20Combinatorics)</sup>

## Proofs and interpretations

There are many routes to the equations. They can be proved by double counting, as in Grünbaum's treatment, but for simplicial polytopes the h-numbers have a natural interpretation in terms of shellings, an ordering of the facets with favorable intersection properties, which yields a compact proof.<sup>[12](https://www.researchgate.net/profile/Fabrice-Clerot/post/Could_anyone_tell_me_what_the_mathematical_definition_expression_for_a_polyhedral_convex_set_and_polyhedral_cone_in_general_Euclidean_space_is/attachment/59d61db679197b80779797ea/AS%3A272440477847554%401441966280124/download/Gallier%2B-%2BConvex%2Bsets%2Betc.pdf)</sup> In the theory of toric varieties, the relations are a manifestation of [Poincaré duality](https://www.edgepedia.org/poincare-duality) for the rational cohomology of the toric variety X_P attached to a simple polytope P; in Cox, Little, and Schenck's account of lattice polytopes, the symmetry h_p = h_{n−p} of the h-numbers is derived from the functional equation of the p-th Ehrhart polynomials and explicitly called the Dehn–Sommerville equations.<sup>[13](https://arxiv.org/abs/1805.01504)</sup><sup> • </sup><sup>[14](https://citeseerx.ist.psu.edu/document?doi=da3c7b13a2c7ba01a9d43cb2a87c55a9051d36ee&repid=rep1&type=pdf)</sup> Matthias Beck's treatment unifies the relations with Ehrhart–Macdonald reciprocity, deriving both from a single reciprocity theorem for face-lattice point enumerators.<sup>[8](https://matthbeck.github.io/papers/ccdnew.pdf)</sup>

## Generalizations

The equations extend well beyond simplicial polytopes. [Victor Klee](https://www.edgepedia.org/victor-klee) defined Eulerian and semi-Eulerian simplicial complexes in 1964 and proved that the h-vector of a (d−1)-dimensional Eulerian complex, such as a simplicial sphere, satisfies h_i = h_{d−i} for all i, while a semi-Eulerian complex, such as the boundary of a simplicial manifold, satisfies the modified relation h_{d−i} = h_i + (−1)^i C(d, i)[χ̃(Γ) − (−1)^{d−1}], where χ̃ is the reduced [Euler characteristic](https://www.edgepedia.org/euler-characteristic).<sup>[15](https://arxiv.org/abs/2003.00160)</sup> Margaret Bayer and Louis Billera characterized in 1985 all linear equalities holding for the flag f-vectors of all Eulerian posets of rank n+1, the generalized Dehn–Sommerville equations; the subspace they determine has dimension given by the [Fibonacci](https://www.edgepedia.org/fibonacci) number e_n with e_0 = e_1 = 1 and e_n = e_{n−1} + e_{n−2}.<sup>[16](https://www.researchgate.net/publication/2142305_Flag_Vectors_of_Eulerian_Partially_Ordered_Sets)</sup> H-vectors of Gorenstein* simplicial posets, a class strictly larger than the boundaries of simplicial convex polytopes, also satisfy the Dehn–Sommerville equations with non-negative components.<sup>[17](https://www.sciencedirect.com/science/article/pii/S0001870804001963)</sup>

More recent work relaxes the Eulerian hypothesis itself. Connor Sawaske and [Lei Xue](https://www.edgepedia.org/lei-xue) established in 2021 three generalizations, covering the h-vectors of pure simplicial complexes, the flag h-vectors of balanced simplicial complexes and graded posets, and the toric h-vectors of graded posets with restricted singularities; in each case any failure of symmetry is expressed in terms of errors coming from the links of faces, extending Klee's semi-Eulerian relations.<sup>[18](https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12072)</sup> Other extensions include Dehn–Sommerville type relations for level sets of the depth function in arrangements of great-spheres, which specialize to the classical relations for simple polytopes at the 0-level,<sup>[11](https://pmc.ncbi.nlm.nih.gov/articles/PMC11415421/)</sup> and a weighted lattice-point generating function whose functional equation simultaneously generalizes Ehrhart–Macdonald reciprocity and the Dehn–Sommerville relations.<sup>[13](https://arxiv.org/abs/1805.01504)</sup> The symmetry of the h-polynomial also permits the expansion defining the gamma vector.<sup>[1](https://mathworld.wolfram.com/Dehn-SommervilleEquations.html)</sup>

## References

1. [Weisstein, Eric W. "Dehn-Sommerville Equations." MathWorld--A Wolfram Resource.](https://mathworld.wolfram.com/Dehn-SommervilleEquations.html)
2. [Polytopes Course Notes, University of Kentucky (MA 714), Chapter 7: The Dehn-Sommerville Equations.](https://www.ms.uky.edu/~Lee/ma714fa13/notes.pdf)
3. [O'Neill, M. Project notes: The Dehn-Sommerville relations, San Diego State University.](https://cdoneill.sdsu.edu/teaching/s22-621/projects.pdf)
4. [McMullen, P. "The maximum numbers of faces of a convex polytope." Mathematika 17 (1970).](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/13483049ECEDF0F2D760C50BAE8D6F21/S0025579300002850a.pdf/div-class-title-the-maximum-numbers-of-faces-of-a-convex-polytope-div.pdf)
5. [Buchstaber, V. M. "Torus actions, combinatorial topology and homological algebra," Russian Mathematical Surveys.](https://www.mathnet.ru/php/getFT.phtml?jrnid=rm&option_lang=eng&paperid=320&what=fullteng)
6. [Brønsted, A. An Introduction to Convex Polytopes, §17: The Dehn–Sommerville Relations.](https://users.fmf.uni-lj.si/lavric/brondsted.pdf)
7. [Moment-angle complexes and combinatorics of simplicial manifolds, ResearchGate.](https://www.researchgate.net/publication/2095872_Moment-angle_complexes_and_combinatorics_of_simplicial_manifolds)
8. [Beck, M. and Pixton, D. Computing the Continuous Discretely, Chapter 5: Face Numbers and the Dehn–Sommerville Relations in Ehrhartian Terms.](https://matthbeck.github.io/papers/ccdnew.pdf)
9. [The Dehn-Sommerville Relations and the Catalan Matroid, arXiv:1512.04513.](https://arxiv.org/abs/1512.04513)
10. [Geometric Combinatorics, Androma reference page.](https://androma.org/page/Geometric%20Combinatorics)
11. [Depth in arrangements: Dehn–Sommerville–Euler relations with applications, PMC11415421.](https://pmc.ncbi.nlm.nih.gov/articles/PMC11415421/)
12. [Gallier, J. Convex sets etc., §7.3: Dehn-Sommerville equations for simplicial polytopes.](https://www.researchgate.net/profile/Fabrice-Clerot/post/Could_anyone_tell_me_what_the_mathematical_definition_expression_for_a_polyhedral_convex_set_and_polyhedral_cone_in_general_Euclidean_space_is/attachment/59d61db679197b80779797ea/AS%3A272440477847554%401441966280124/download/Gallier%2B-%2BConvex%2Bsets%2Betc.pdf)
13. [Weighted lattice point sums in lattice polytopes, unifying Dehn–Sommerville and Ehrhart–Macdonald, arXiv:1805.01504.](https://arxiv.org/abs/1805.01504)
14. [Cox, D., Little, J., Schenck, H. Toric Varieties, §9.4: Applications to Lattice Polytopes.](https://citeseerx.ist.psu.edu/document?doi=da3c7b13a2c7ba01a9d43cb2a87c55a9051d36ee&repid=rep1&type=pdf)
15. [Non-Eulerian Dehn-Sommerville relations, arXiv:2003.00160.](https://arxiv.org/abs/2003.00160)
16. [Flag Vectors of Eulerian Partially Ordered Sets, ResearchGate.](https://www.researchgate.net/publication/2142305_Flag_Vectors_of_Eulerian_Partially_Ordered_Sets)
17. [Masuda, M. "h-Vectors of Gorenstein* simplicial posets." Advances in Mathematics 194 (2005) 332–344.](https://www.sciencedirect.com/science/article/pii/S0001870804001963)
18. [Sawaske, C. and Xue, L. "Non-Eulerian Dehn–Sommerville relations." Mathematika 67 (2021) 257–287.](https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.12072)

---
*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Logic and discrete mathematics › General discrete mathematics and discrete structures › Combinatorics › Geometric, polyhedral, and topological combinatorics › Face numbers and face vectors*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license