Locally Anti-Blocking Polytopes

• Locally anti-blocking polytopes are convex bodies whose structure is anti-blocking in every orthant, satisfying the coordinate section–projection identity.
• They exhibit remarkable geometric stability under coordinate projections and duality, leading to sharp results in mixed-volume theory and face enumeration.
• Research on these polytopes reveals deep connections with Ehrhart theory, L_p-Rogers–Shephard inequalities, and flow polytope realizations, unifying combinatorial and geometric insights.

Searching arXiv for recent and foundational papers on locally anti-blocking polytopes. arXiv search query: locally anti-blocking polytopes Locally anti-blocking polytopes are convex polytopes whose geometry is anti-blocking in every orthant. In the most common formulation, a polytope $P\subset \mathbb R^d$ is locally anti-blocking if, for every coordinate subspace $H$, the orthogonal projection onto $H$ coincides with the coordinate section: $\operatorname{proj}_H(P)=P\cap H.$ Equivalently, in each orthant the corresponding piece of $P$ becomes an anti-blocking polytope after the appropriate sign changes. This class contains anti-blocking polytopes in $\mathbb R^d_{\ge 0}$, contains all unconditional polytopes, and has become a natural setting for sharp results on mixed volumes, face enumeration, Ehrhart theory, and combinatorial realizations via flow polytopes (Sadovsky, 2023, Sanyal et al., 2023, Ohsugi et al., 2019, Berggren et al., 26 May 2026).

1. Definition and equivalent formulations

An anti-blocking polytope in the nonnegative orthant is a convex polytope $P\subset \mathbb R^d_{\ge 0}$ such that

$$x\in P,\quad 0\le y\le x \;\Longrightarrow\; y\in P.$$

Thus $P$ is downward closed with respect to the coordinatewise order. A polytope is locally anti-blocking if this property holds orthant by orthant: for every sign vector $\sigma\in\{\pm 1\}^d$, the orthant piece

$H$0

is anti-blocking after translating it back to the positive orthant by the sign-change map $H$1 (Sadovsky, 2023).

A fundamental equivalent characterization is the coordinate section–projection identity. For every coordinate subspace $H$2,

$H$3

This criterion appears repeatedly because it converts orthantwise order-convexity into an intrinsic geometric condition and is the form most directly used in later inequalities and inductive arguments (Sanyal et al., 2023).

In the lattice setting, one often phrases the definition orthantwise up to unimodular equivalence. A full-dimensional lattice polytope $H$4 is locally anti-blocking if for each closed orthant $H$5 there exists an anti-blocking lattice polytope $H$6 such that $H$7 is unimodularly equivalent to $H$8 by reflections in coordinate hyperplanes (Ohsugi et al., 2019).

This class properly extends unconditionality. A $H$9-unconditional body is exactly a locally anti-blocking body for which all orthant pieces are congruent to a fixed anti-blocking body in $H$0 (Sadovsky, 2023). Conversely, local anti-blocking does not force central symmetry: for example,

$H$1

is locally anti-blocking but not centrally symmetric (Chor, 29 Jul 2025).

2. Structural geometry

For proper locally anti-blocking polytopes, meaning $H$2, the section–projection identity is stable under passing to coordinate subspaces. If

$H$3

then

$H$4

and each $H$5 is again locally anti-blocking (Sanyal et al., 2023). This recursive closure is central in proofs by dimension reduction.

Duality is equally rigid. The polar $H$6 of a locally anti-blocking polytope is again locally anti-blocking, and the restriction to coordinate subspaces commutes with polarity: $H$7 The same source also records a normal-cone restriction: $H$8 for $H$9. These identities make locally anti-blocking geometry unusually compatible with coordinate decompositions (Sanyal et al., 2023).

A complementary characterization uses the normal fan. A full-dimensional polytope $\operatorname{proj}_H(P)=P\cap H.$0 is locally anti-blocking if and only if all facet normals lie in the union of the coordinate orthants and, for each sign vector $\operatorname{proj}_H(P)=P\cap H.$1, the normal cone of the face $\operatorname{proj}_H(P)=P\cap H.$2 is spanned by the coordinate normals $\operatorname{proj}_H(P)=P\cap H.$3 for some $\operatorname{proj}_H(P)=P\cap H.$4. Equivalently, the normal fan of $\operatorname{proj}_H(P)=P\cap H.$5 refines the arrangement of coordinate hyperplanes (Artstein-Avidan et al., 2020).

These structural restrictions imply a rigid local form for facets. In particular, if $\operatorname{proj}_H(P)=P\cap H.$6 is proper, then the unique facet whose relative interior lies in the positive orthant is a simplex

$\operatorname{proj}_H(P)=P\cap H.$7

This simplex-like behavior of orthant facets is one reason simplices recur as equality cases in volume inequalities, even though Hanner polytopes control extremal face counts (Sanyal et al., 2023).

3. Mixed-volume theory and Godbersen-type inequalities

The mixed-volume theory of locally anti-blocking polytopes is built from orthantwise decompositions. If $\operatorname{proj}_H(P)=P\cap H.$8 is a locally anti-blocking body and $\operatorname{proj}_H(P)=P\cap H.$9, then

$P$0

This reduces the global mixed volume to a sum of anti-blocking contributions (Sadovsky, 2023).

For anti-blocking polytopes $P$1, the basic geometric decomposition is

$P$2

with disjoint interiors. From this one obtains the mixed-volume formula

$P$3

The same framework also yields the reverse Kleitman inequality

$P$4

and in particular $P$5 (Artstein-Avidan et al., 2020).

Within this setting, Godbersen’s conjecture is known to hold. For every locally anti-blocking body $P$6 and every $P$7,

$P$8

The proof combines four ingredients: orthant decomposition of mixed volume, a mixed-volume reverse-Kleitman step in opposite orthants, the coordinate-projection formula for anti-blocking mixed volumes, and the classical Rogers–Shephard inequality applied to coordinate sections and projections (Sadovsky, 2023).

The equality theory is especially rigid. The only places where strictness can occur are the reverse-Kleitman step and the Rogers–Shephard step. Equality in those forces the orthant pieces to be coordinate simplices of the form

$P$9

A key calculation for aligned simplices is

$\mathbb R^d_{\ge 0}$0

where $\mathbb R^d_{\ge 0}$1 and $\mathbb R^d_{\ge 0}$2. Comparing opposite orthants then shows that equality in Godbersen’s inequality occurs only when $\mathbb R^d_{\ge 0}$3 itself is a simplex (Sadovsky, 2023).

4. Face enumeration and extremal combinatorics

Locally anti-blocking polytopes also satisfy strong lower bounds on the number of faces. For a proper locally anti-blocking $\mathbb R^d_{\ge 0}$4-polytope $\mathbb R^d_{\ge 0}$5, the number $\mathbb R^d_{\ge 0}$6 of nonempty faces satisfies

$\mathbb R^d_{\ge 0}$7

The proof assigns to each sign vector $\mathbb R^d_{\ge 0}$8 a strictly concave function

$\mathbb R^d_{\ge 0}$9

on the open cone $P\subset \mathbb R^d_{\ge 0}$0, then uses the unique maximizer $P\subset \mathbb R^d_{\ge 0}$1 to define a special face $P\subset \mathbb R^d_{\ge 0}$2. Distinct sign vectors give distinct faces, producing $P\subset \mathbb R^d_{\ge 0}$3 faces in total (Sanyal et al., 2023).

The equality case is governed not by simplices but by Hanner structure. Equality $P\subset \mathbb R^d_{\ge 0}$4 holds only for generalized Hanner polytopes. The argument passes to all coordinate sections, shows that every $P\subset \mathbb R^d_{\ge 0}$5-dimensional coordinate section is either a rectangle or a diamond, and encodes the resulting pattern by a graph $P\subset \mathbb R^d_{\ge 0}$6 on $P\subset \mathbb R^d_{\ge 0}$7, with $P\subset \mathbb R^d_{\ge 0}$8 an edge exactly when the corresponding $P\subset \mathbb R^d_{\ge 0}$9-plane section is axis-aligned. The extremal graph must be a cograph, and cographs are precisely the recursive combinatorial skeleton underlying the direct product and direct sum operations that generate Hanner polytopes (Sanyal et al., 2023).

A parallel phenomenon holds for complete flags. If $$x\in P,\quad 0\le y\le x \;\Longrightarrow\; y\in P.$$0 is a normalized locally anti-blocking polytope, meaning

$$x\in P,\quad 0\le y\le x \;\Longrightarrow\; y\in P.$$1

then

$$x\in P,\quad 0\le y\le x \;\Longrightarrow\; y\in P.$$2

The proof decomposes full flags by their orthant sign and constructs, for each full-dimensional cone $$x\in P,\quad 0\le y\le x \;\Longrightarrow\; y\in P.$$3 in the orthant fan and each facet $$x\in P,\quad 0\le y\le x \;\Longrightarrow\; y\in P.$$4, an injective map

$$x\in P,\quad 0\le y\le x \;\Longrightarrow\; y\in P.$$5

Induction on the dimension of $$x\in P,\quad 0\le y\le x \;\Longrightarrow\; y\in P.$$6 yields $$x\in P,\quad 0\le y\le x \;\Longrightarrow\; y\in P.$$7, and summing over all $$x\in P,\quad 0\le y\le x \;\Longrightarrow\; y\in P.$$8 orthants gives the stated lower bound (Chor, 29 Jul 2025).

Again the equality case is Hanner. If $$x\in P,\quad 0\le y\le x \;\Longrightarrow\; y\in P.$$9, then every coordinate section $P$0 also attains the bound, each $P$1-coordinate section is either $P$2 or $P$3, and $P$4 is reconstructed from the graph $P$5. The resulting graph contains no induced path on $P$6 vertices, hence is a cograph. In this class, equality holds if and only if $P$7 is a generalized Hanner polytope (Chor, 29 Jul 2025).

5. Lattice structure, $P$8-polynomials, and $P$9-positivity

For lattice polytopes, the locally anti-blocking condition admits a direct Ehrhart-theoretic decomposition. Let $\sigma\in\{\pm 1\}^d$0 be a locally anti-blocking lattice polytope of dimension $\sigma\in\{\pm 1\}^d$1, and let $\sigma\in\{\pm 1\}^d$2 be anti-blocking representatives of the orthant pieces. If $\sigma\in\{\pm 1\}^d$3 denotes the unconditional polytope obtained by reflecting $\sigma\in\{\pm 1\}^d$4 in all coordinate hyperplanes, then the $\sigma\in\{\pm 1\}^d$5-polynomial satisfies

$\sigma\in\{\pm 1\}^d$6

This formula expresses the Ehrhart series of a locally anti-blocking lattice polytope as the average of the $\sigma\in\{\pm 1\}^d$7-polynomials of unconditional polytopes built from its orthant pieces (Ohsugi et al., 2019).

The proof proceeds through an inclusion–exclusion formula for an anti-blocking polytope $\sigma\in\{\pm 1\}^d$8: $\sigma\in\{\pm 1\}^d$9 Applying this orthant by orthant and recombining the contributions yields the average formula above (Ohsugi et al., 2019).

This has an immediate consequence for reflexive locally anti-blocking polytopes. If $H$00 is reflexive and every unconditional polytope $H$01 has a $H$02-positive $H$03-polynomial, then $H$04 is also $H$05-positive. The reason is that reflexivity makes these $H$06-polynomials palindromic, and averages of $H$07-positive palindromic polynomials remain $H$08-positive (Ohsugi et al., 2019).

The worked examples in the same source place several familiar families inside this framework. Every pseudo-symmetric simplicial reflexive polytope, being a free sum of copies of the cross-polytope, the del Pezzo polytope, and the pseudo-del Pezzo polytope, has $H$09-positive $H$10-polynomial. The symmetric edge polytope $H$11 of a cycle also has

$H$12

so its $H$13-vector is explicitly nonnegative (Ohsugi et al., 2019).

6. Flow-polytopal realizations and further extremal inequalities

A recent combinatorial realization of local anti-blocking behavior arises from flow polytopes. For an acyclic directed graph $H$14 with source $H$15 and sink $H$16, the strength-one flow polytope is

$H$17

If $H$18 admits an ample framing $H$19, then $H$20 is Gorenstein and projects along a distinguished special simplex to a reflexive polytope $H$21, the $H$22-polytope (Berggren et al., 26 May 2026).

In this setting, local anti-blocking can be characterized combinatorially. A connected full DAG $H$23 admits an ample framing $H$24 such that $H$25 is locally anti-blocking if and only if the inner graph of $H$26 is either a path or a cycle. These families are denoted $H$27 and $H$28 (Berggren et al., 26 May 2026).

For such $H$29-polytopes, the face structure can be read from routes. A set $H$30 lies in a common face if and only if the union $H$31 of the corresponding edge sets does not contain all edges of any exceptional route. In particular, if two routes $H$32 are coherent, then the minimal face containing them is an edge, while for routes in conflict the minimal face is a quadrilateral unless an exceptional route intervenes, in which case the minimal face is the whole polytope (Berggren et al., 26 May 2026). The same paper proves that the DKK triangulation of $H$33 projects to a regular unimodular triangulation of $H$34, and in the locally anti-blocking path-or-cycle case this triangulation is precisely a pulling triangulation described by coherence diagrams.

Another major line concerns $H$35-Rogers–Shephard inequalities. For locally anti-blocking bodies $H$36 and $H$37,

$H$38

where $H$39 is Firey’s $H$40-sum. For $H$41, the equality cases are completely characterized: $H$42 for nonzero scalars $H$43 (Fradelizi et al., 5 Jun 2026). For polytopes, this means that the only extremizers are coordinate simplices with one vertex at the origin.

These results show that extremal behavior in the locally anti-blocking class depends strongly on the functional under study. Mixed-volume and $H$44-Rogers–Shephard inequalities isolate simplices as the unique equality cases (Sadovsky, 2023, Fradelizi et al., 5 Jun 2026), whereas global face-minimization problems isolate generalized Hanner polytopes (Sanyal et al., 2023, Chor, 29 Jul 2025). This distinction is one of the central organizing features of the modern theory of locally anti-blocking polytopes.