Secondary Polytopes

• Secondary polytopes are convex polytopes derived from finite point configurations that capture all regular triangulations via GKZ vectors.
• They structure and encode regular subdivisions, enabling face factorization and revealing deep links between flip combinatorics and toric geometry.
• Applications span toric moduli, tropical computation, and higher secondary generalizations like associahedra and plabic graphs in combinatorial geometry.

Secondary polytopes are the convex polytopes attached to finite point configurations $A\subset \mathbb R^d$ that organize regular triangulations and, more generally, regular polyhedral subdivisions of $\operatorname{Conv}(A)$. A height vector induces a regular subdivision by lifting $a_i\mapsto (a_i,\omega_i)$, taking the lower hull, and projecting back; the resulting secondary fan is the normal fan of the secondary polytope $\Sigma(A)$. Its vertices correspond to regular triangulations, its lower-dimensional faces correspond to coarser regular subdivisions, and its dimension is $|A|-d-1$ in the standard notation, equivalently $N-n$ when $A=\{a_0,\dots,a_N\}\subset \mathbb Z^n$ (Yotsutani, 2013).

1. Definitions, GKZ vectors, and fiber-polytope realizations

Let $A=\{a_1,\dots,a_m\}\subset \mathbb R^d$ affinely span $\mathbb R^d$, and write $Q=\operatorname{Conv}(A)$. A triangulation $\operatorname{Conv}(A)$0 of $\operatorname{Conv}(A)$1 is a subdivision of $\operatorname{Conv}(A)$2 into $\operatorname{Conv}(A)$3-simplices with vertices in $\operatorname{Conv}(A)$4. Regularity is imposed by a lifting construction: one chooses a height or weight vector $\operatorname{Conv}(A)$5, lifts $\operatorname{Conv}(A)$6, and projects the lower convex hull back to $\operatorname{Conv}(A)$7. If the cells are simplices, the result is a regular triangulation; if not, one obtains a regular subdivision (Yotsutani, 2013).

To a triangulation one associates a GKZ vector, written in the literature as $\operatorname{Conv}(A)$8, $\operatorname{Conv}(A)$9, $a_i\mapsto (a_i,\omega_i)$0, or $a_i\mapsto (a_i,\omega_i)$1. In one standard normalization,

$a_i\mapsto (a_i,\omega_i)$2

while an equivalent formulation, up to the chosen volume normalization, is

$a_i\mapsto (a_i,\omega_i)$3

The secondary polytope is then defined as the convex hull of these vectors; some treatments take the convex hull over all triangulations and then prove that the vertices are exactly the regular ones, while others define it directly as the convex hull over regular triangulations (Yotsutani, 2013).

A complementary realization is the Billera–Sturmfels fiber-polytope description. If

$a_i\mapsto (a_i,\omega_i)$4

then $a_i\mapsto (a_i,\omega_i)$5 is the fiber polytope, equivalently the Minkowski integral over the fibers of $a_i\mapsto (a_i,\omega_i)$6. Its support function admits the discrete formula

$a_i\mapsto (a_i,\omega_i)$7

and in the planar convex-position case the resulting polytope is the associahedron (Esterov et al., 2024).

2. Secondary fans, face lattices, and factorization

The space of height vectors is partitioned by the regular subdivisions they induce. These polyhedral cones form the secondary fan, and the normal fan of $a_i\mapsto (a_i,\omega_i)$8 coincides with this fan. The basic order-theoretic statement is that the face lattice of the secondary polytope is anti-isomorphic to the poset of regular subdivisions ordered by refinement (Kapranov et al., 2014).

For a regular subdivision $a_i\mapsto (a_i,\omega_i)$9, the corresponding face is

$\Sigma(A)$0

The inclusion relation is order-reversing: $\Sigma(A)$1 In particular, vertices correspond to regular triangulations, while facets correspond to coarse subdivisions, namely regular subdivisions strictly finer than the trivial one but with no intermediate refinement (Yotsutani, 2013).

A central structural property is factorization. If a regular subdivision $\Sigma(A)$2 decomposes $\Sigma(A)$3 into cells $\Sigma(A)$4 with $\Sigma(A)$5, then

$\Sigma(A)$6

This product decomposition is the geometric content behind many later algebraic and categorical constructions (Kapranov et al., 17 Sep 2025).

Facet equations can be written explicitly. If a coarse subdivision $\Sigma(A)$7 is induced by a height vector $\Sigma(A)$8, then the supporting inequality for the corresponding facet is

$\Sigma(A)$9

with equality on that facet. In another standard formulation, circuits of $|A|-d-1$0 determine the facet inequalities cutting off facets of $|A|-d-1$1 (Yotsutani, 2013).

A recurrent misconception is that every triangulation of $|A|-d-1$2 should determine a vertex. The precise statement is narrower: the vertices are exactly the regular triangulations. Non-regular triangulations may enter the ambient combinatorics of $|A|-d-1$3, but not the vertex set of $|A|-d-1$4.

3. Planar configurations, associahedra, and extremal $|A|-d-1$5-vectors

For $|A|-d-1$6 points in general position in the plane, the secondary polytope has dimension $|A|-d-1$7. If $|A|-d-1$8 is a convex $|A|-d-1$9-gon, $N-n$0 is the classical $N-n$1-dimensional associahedron. Writing $N-n$2, its face numbers admit the closed form

$N-n$3

In particular,

$N-n$4

the usual Catalan number $N-n$5 (Fernández et al., 2021).

The sharp planar extremal theorem states that if $N-n$6 is any $N-n$7-point planar configuration in general position and $N-n$8 is $N-n$9 points in convex position, then for each $A=\{a_0,\dots,a_N\}\subset \mathbb Z^n$0,

$A=\{a_0,\dots,a_N\}\subset \mathbb Z^n$1

Thus convex position minimizes the entire $A=\{a_0,\dots,a_N\}\subset \mathbb Z^n$2-vector among all planar configurations. In the language of the paper, convex position not only maximizes the number of triangulations, it actually minimizes the entire $A=\{a_0,\dots,a_N\}\subset \mathbb Z^n$3-vector among all planar configurations, and no “more concave” planar set can have strictly fewer faces in its secondary polytope than the associahedron (Fernández et al., 2021).

The proof strategy uses a boundary vertex $A=\{a_0,\dots,a_N\}\subset \mathbb Z^n$4, a monotone-link signature $A=\{a_0,\dots,a_N\}\subset \mathbb Z^n$5, and an associated base subdivision called the extended star. A key lemma shows that this extended star is itself regular, so refinements obtained by further subdividing each cell remain regular after a small perturbation of the realizing height vector. Counting is then reduced to Catalan-type product formulas on negative intervals together with inequalities such as

$A=\{a_0,\dots,a_N\}\subset \mathbb Z^n$6

and their refinements for face numbers (Fernández et al., 2021).

For the hexagon, the associahedron has

$A=\{a_0,\dots,a_N\}\subset \mathbb Z^n$7

Accordingly, every $A=\{a_0,\dots,a_N\}\subset \mathbb Z^n$8-point set in general position has at least $A=\{a_0,\dots,a_N\}\subset \mathbb Z^n$9 vertices, $A=\{a_1,\dots,a_m\}\subset \mathbb R^d$0 edges, $A=\{a_1,\dots,a_m\}\subset \mathbb R^d$1 two-faces, and $A=\{a_1,\dots,a_m\}\subset \mathbb R^d$2 three-face in its secondary polytope (Fernández et al., 2021).

4. Toric geometry, Chow stability, and moduli interpretations

Secondary polytopes occupy a central position in toric geometry. For a projective toric variety $A=\{a_1,\dots,a_m\}\subset \mathbb R^d$3, Kapranov–Sturmfels–Zelevinsky identify the Chow polytope with the secondary polytope: $A=\{a_1,\dots,a_m\}\subset \mathbb R^d$4 Under the Hilbert–Mumford criterion for a torus action, semistability is equivalent to $A=\{a_1,\dots,a_m\}\subset \mathbb R^d$5, while polystability is equivalent to $A=\{a_1,\dots,a_m\}\subset \mathbb R^d$6 lying in the interior of $A=\{a_1,\dots,a_m\}\subset \mathbb R^d$7. A convex-geometric argument via facet equations then proves that Chow semistability is equivalent to Chow polystability for projective toric varieties (Yotsutani, 2013).

The facet description is operational in this setting. One enumerates coarse regular subdivisions, records their height vectors $A=\{a_1,\dots,a_m\}\subset \mathbb R^d$8, computes the corresponding right-hand sides $A=\{a_1,\dots,a_m\}\subset \mathbb R^d$9, and obtains a complete facet presentation

$\mathbb R^d$0

From this description one reads off whether $\mathbb R^d$1 lies in the interior or on the boundary (Yotsutani, 2013).

A parallel viewpoint arises from irrational toric geometry. For a general real configuration $\mathbb R^d$2, one builds an irrational toric variety associated to the secondary fan, and the space of Hausdorff limits of translates of the irrational projective toric model $\mathbb R^d$3 is canonically homeomorphic to the irrational toric variety $\mathbb R^d$4. When the fan is the normal fan of a polytope, the irrational toric variety is homeomorphic to that polytope; in the rational case the nonnegative real part of the secondary toric variety is identified with the secondary polytope by a moment-type map (Pir et al., 2018).

These interpretations explain why secondary polytopes recur in the study of generalized discriminants, resultants, and GKZ hypergeometric systems. A plausible implication is that the secondary polytope is best viewed not merely as a combinatorial hull of triangulation data, but as a universal toric object controlling degenerations indexed by regular subdivisions.

5. Higher secondary polytopes and generalizations

Galashin, Postnikov, and Williams introduced higher secondary polytopes $\mathbb R^d$5, with $\mathbb R^d$6 equal to the classical GKZ secondary polytope. Their Minkowski sum agrees with the Billera–Sturmfels fiber zonotope associated with a lift of $\mathbb R^d$7, and for each $\mathbb R^d$8 one has

$\mathbb R^d$9

Vertices of $Q=\operatorname{Conv}(A)$0 correspond to $Q=\operatorname{Conv}(A)$1-equivalence classes of regular tilings, and edges correspond to flips of level $Q=\operatorname{Conv}(A)$2 (Galashin et al., 2019).

In the case $Q=\operatorname{Conv}(A)$3, these are higher associahedra. When $Q=\operatorname{Conv}(A)$4 is the vertex set of a convex $Q=\operatorname{Conv}(A)$5-gon, the vertices of $Q=\operatorname{Conv}(A)$6 correspond to regular bipartite $Q=\operatorname{Conv}(A)$7-plabic graphs, and the edges correspond to square moves. For $Q=\operatorname{Conv}(A)$8 and $Q=\operatorname{Conv}(A)$9, one obtains a $\operatorname{Conv}(A)$00-dimensional polytope with $\operatorname{Conv}(A)$01 vertices and diameter $\operatorname{Conv}(A)$02 (Galashin et al., 2019).

Bullock and Gravel study a closely related family of higher secondary polytopes for two-dimensional zonotopes. In their planar zonotopal setting, the $\operatorname{Conv}(A)$03-skeleton of $\operatorname{Conv}(A)$04 is isomorphic to the flip graph of $\operatorname{Conv}(A)$05-hypertriangulations, and the diameters of $\operatorname{Conv}(A)$06 and $\operatorname{Conv}(A)$07 admit explicit closed-form expressions depending only on $\operatorname{Conv}(A)$08 and $\operatorname{Conv}(A)$09 (Bullock et al., 2020).

A different extension is the framework of basecondary polytopes. This class contains classical secondary polytopes, submodular base-polytopes, Minkowski sums of secondary polytopes and submodular base-polytopes, and Newton polytopes of discriminants and resultants. In this language, the classical secondary polytope appears as a special case of a more general support-function formalism (Esterov et al., 2024).

These generalizations preserve a common pattern: a polytope whose normal fan refines the secondary fan, whose vertices encode coherent combinatorial data, and whose edges record elementary flip operations. The classical GKZ polytope is the first instance of a broader hierarchy rather than an isolated construction.

6. Homotopical, categorical, computational, and recent combinatorial directions

The factorization of faces of $\operatorname{Conv}(A)$10 supports substantial homotopical algebra. Chains on secondary polytopes assemble into a commutative dg-algebra, and by Koszul duality one obtains an $\operatorname{Conv}(A)$11-algebra $\operatorname{Conv}(A)$12. In dimension $\operatorname{Conv}(A)$13, Kapranov and Kontsevich show that the resulting $\operatorname{Conv}(A)$14-structure is quasi-isomorphic to the directed Hochschild complex of a strictly triangular $\operatorname{Conv}(A)$15-algebra $\operatorname{Conv}(A)$16; in the language of the Algebra of the Infrared, the convex-geometric data of secondary polytopes reproduces the $\operatorname{Conv}(A)$17- or $\operatorname{Conv}(A)$18-algebra of plane webs (Kapranov et al., 2014).

The same geometry reappears in the passage from perverse sheaves to perverse schobers. For a finite set of singularities $\operatorname{Conv}(A)$19, rectilinear transport maps satisfy convex-path-sum identities governed by the secondary-polytope fan. Kapranov–Schechtman’s categorification replaces vanishing-cycle vector spaces by triangulated categories and transport maps by functors; the resulting $\operatorname{Conv}(A)$20-monad $\operatorname{Conv}(A)$21 and its Hochschild-type invariants are again organized by secondary-polytope combinatorics (Kapranov et al., 17 Sep 2025).

On the computational side, secondary polytopes arise as special cases of mixed fiber polytopes. The software TrIm implements tropical implicitization and tropical elimination; when the image is a hypersurface, the output is the Newton polytope of the defining polynomial, and TrIm can therefore be used to compute mixed fiber polytopes, including secondary polytopes. The workflow described in the source passes through a tropical complete intersection, projection to the relevant linear quotient, and reconstruction of the Newton polytope from the resulting tropical hypersurface (0706.0564).

Recent work on the hypersimplex $\operatorname{Conv}(A)$22 provides another geometric incarnation of the secondary fan. For the $\operatorname{Conv}(A)$23-action on $\operatorname{Conv}(A)$24, an arrangement $\operatorname{Conv}(A)$25 cuts $\operatorname{Conv}(A)$26 into chambers whose open regions coincide with maximal-dimensional GKZ cones, and the cones over these regions form exactly the secondary fan of the cone spanned by the vertices of $\operatorname{Conv}(A)$27 (Sergeev, 5 Jun 2026).

Taken together, these developments show that secondary polytopes mediate among convex geometry, toric moduli, flip combinatorics, tropical computation, homotopy algebra, and categorical structures. Their persistent role is to convert coherent subdivision data into a polyhedral object whose faces, normals, and factorization properties are simultaneously combinatorial and geometric.