Bounded Intersection Property in Convex Geometry

• Bounded Intersection Property is a condition in convex geometry that requires every bounded face of a halfspace intersection polyhedron to have dimension at most d, thereby controlling its complexity.
• Methodologies involve vertex enumeration via subsets of at most d supporting hyperplanes, yielding polynomial bounds such as O(n^d) for vertex counts.
• General position assumptions refine the face count from O(n^(d^2)) to O(n^d), enabling more efficient algorithms for constructing the bounded subcomplex.

Searching arXiv for the primary paper and closely related uses of “intersection property” to ground terminology and citations. In the convex-geometric setting of halfspace intersections, the “bounded intersection property” (Editor’s term) denotes the condition that a polyhedron $P=\bigcap_{i=1}^n H_i\subset \mathbb{R}^D$ may be unbounded in many directions, but every bounded face of $P$ has dimension at most $d$. In the paper "Bounds on the Complexity of Halfspace Intersections when the Bounded Faces have Small Dimension" (Eppstein et al., 2011), this condition is expressed equivalently by requiring that the bounded subcomplex $P^{\ell<\infty}$, or more generally the thresholded subcomplex $P^{\ell<B}$, has dimension $d<D$. The resulting theory shows that the combinatorial and algorithmic complexity of the bounded part of a high-dimensional halfspace intersection depends polynomially on $n$ with exponents governed by $d$, not by the ambient dimension $D$.

1. Formal setting and equivalent formulations

Let

$P=\bigcap_{i=1}^n H_i,$

where each $P$0 is a closed $P$1-dimensional halfspace. A face of $P$2 is any set of the form $P$3, where $P$4 is a closed halfspace whose boundary hyperplane does not intersect the relative interior of $P$5. Faces include the empty set of dimension $P$6, all proper faces, and possibly $P$7 itself. A face is bounded if it is bounded in the Euclidean sense; vertices are $P$8-dimensional faces, bounded edges are $P$9-faces, while rays and lines are unbounded $d$0-faces (Eppstein et al., 2011).

The central parameter is the maximum dimension of a bounded face. If $d$1 is a linear function attaining a minimum on $d$2, and $d$3, the paper considers

$d$4

When $d$5, this is exactly the bounded subcomplex $d$6. The bounded intersection property is then the requirement

$d$7

and in particular, for $d$8,

$d$9

This formulation isolates the bounded geometry of an otherwise unbounded polyhedron. The ambient polyhedron may have extensive unbounded structure, but every bounded connected piece is confined to a $P^{\ell<\infty}$0-dimensional face structure. This suggests a low-dimensional bounded skeleton embedded in $P^{\ell<\infty}$1.

2. Vertex structure and global combinatorial bounds

The decisive structural lemma states that every vertex $P^{\ell<\infty}$2 of $P^{\ell<\infty}$3 arises as an $P^{\ell<\infty}$4-minimum on an affine subspace cut out by at most $P^{\ell<\infty}$5 supporting hyperplanes: there exists a set $P^{\ell<\infty}$6 of at most $P^{\ell<\infty}$7 defining halfspaces such that, if $P^{\ell<\infty}$8 is the intersection of the boundary hyperplanes of $P^{\ell<\infty}$9, then

$P^{\ell<B}$0

Thus bounded vertices are controlled by at most $P^{\ell<B}$1 inequalities rather than $P^{\ell<B}$2 inequalities (Eppstein et al., 2011).

From this, the paper derives an exact upper bound for vertices. If $P^{\ell<B}$3 is a full-dimensional pointed polyhedron, $P^{\ell<B}$4 is greater than the minimum of $P^{\ell<B}$5, and $P^{\ell<B}$6, then the number of vertices of $P^{\ell<B}$7 is at most

$P^{\ell<B}$8

Asymptotically, for fixed $P^{\ell<B}$9,

$d<D$0

with constant independent of $d<D$1.

The same parameter $d<D$2 controls the number of bounded faces. Without any general position assumption, the total number of faces in $d<D$3 is

$d<D$4

The argument uses two ingredients. First, every face of dimension at most $d<D$5 is the intersection of the affine hull of a set of at most $d<D$6 vertices, and there are $d<D$7 vertices. Second, an Euler-characteristic identity shows that when $d<D$8 is nonempty,

$d<D$9

so the number of $n$0-faces is no larger than the total number of lower-dimensional faces.

Quantity	Bound
Vertices of $n$1	$n$2
Vertices, fixed $n$3	$n$4
Total faces, arbitrary position	$n$5

These estimates replace classical worst-case dependence on $n$6 by dependence on the bounded-face dimension $n$7. The data explicitly notes that the bounds are tight in the exponent for the vertex count, and for some small values of $n$8 also for faces.

3. General position and sharper face counts

A substantially stronger theory holds under general position. The paper defines an intersection of halfspaces to be in general position if a small perturbation of any halfspace does not change the combinatorial structure. In particular, a full-dimensional polyhedron in general position is simple: every vertex is incident to exactly $n$9 facets, and the link of each vertex is a simplex (Eppstein et al., 2011).

If $d$0 is in general position, has $d$1 vertices in total, and $d$2, then for every $d$3, the number of $d$4-dimensional faces of $d$5 is at most

$d$6

The proof charges each $d$7-face to its $d$8-maximum vertex; these faces correspond to $d$9-faces of the link at that vertex, and the link is a simplex of dimension at most $D$0.

Combining this with the vertex estimate $D$1 yields

$D$2

for fixed $D$3. Hence general position collapses the general $D$4 face bound to $D$5.

This refinement is not a routine perturbative consequence. The paper explicitly notes that it is not always possible to perturb an instance into general position without increasing the dimension of the bounded subcomplex, and gives an explicit $D$6-dimensional example in which perturbation increases that dimension from $D$7 to $D$8. Accordingly, the sharper general-position estimates do not automatically transfer to arbitrary instances.

4. Algorithms and polynomial-time construction

Let $D$9 denote the time required to solve a linear program in dimension $P=\bigcap_{i=1}^n H_i,$0 with $P=\bigcap_{i=1}^n H_i,$1 constraints, or to detect infeasibility or unboundedness, in an exact arithmetic model. The algorithmic results are parameterized by the same bounded-face dimension $P=\bigcap_{i=1}^n H_i,$2, and they use the vertex lemma above as their basic primitive (Eppstein et al., 2011).

When $P=\bigcap_{i=1}^n H_i,$3 is known, all vertices of $P=\bigcap_{i=1}^n H_i,$4 can be computed by enumerating all subsets $P=\bigcap_{i=1}^n H_i,$5 with $P=\bigcap_{i=1}^n H_i,$6, intersecting the corresponding boundary hyperplanes to form $P=\bigcap_{i=1}^n H_i,$7, minimizing $P=\bigcap_{i=1}^n H_i,$8 over $P=\bigcap_{i=1}^n H_i,$9, and deduplicating the feasible bounded optima. There are $P$00 such subsets, so the running time is

$P$01

Once the vertices are available, two routes are given for constructing the full bounded subcomplex. In general position, the paper combines vertex enumeration with vertex-facet incidences and the algorithm of Herrmann et al., obtaining

$P$02

Without general position, the same pipeline yields

$P$03

A more direct LP-based enumeration is also described. Starting from the list of vertices, it tests affine hulls generated by a face $P$04 and a vertex $P$05, checking whether the resulting intersection with $P$06 is a new bounded face. This gives

$P$07

in general position and

$P$08

without the general position hypothesis.

When $P$09 is unknown, the paper runs the known-$P$10 constructions for $P$11, and uses a stopping criterion given by Lemma 11, the “no gap” lemma. Roughly, this lemma certifies that once all candidate vertices produced by subsets of at most $P$12 halfspaces have been found, all currently known bounded faces have dimension at most $P$13, and every affine hull generated by a known face and a known vertex either meets the interior of $P$14, is already represented, or is unbounded under $P$15, then the current collection is exactly the bounded subcomplex.

The resulting complexity bounds, with $P$16 the actual maximum dimension of a bounded face, are: $P$17 without general position, and

$P$18

with general position. For fixed $P$19, the number of LP calls and the total running time are polynomial in $P$20 and $P$21.

Task	Assumptions	Time
Enumerate vertices	$P$22 known	$P$23
Build bounded subcomplex	general position	$P$24
Build bounded subcomplex	arbitrary position	$P$25
Direct LP enumeration	general position	$P$26
Direct LP enumeration	arbitrary position	$P$27
Unknown $P$28	general position	$P$29
Unknown $P$30	arbitrary position	$P$31

5. Geometric meaning and relation to broader polyhedral theory

Geometrically, the bounded intersection property does not assert that the whole intersection $P$32 is low-dimensional. Rather, it asserts that every bounded face is low-dimensional. For $P$33, the bounded subcomplex is a finite set of isolated vertices; for $P$34, it is a graph-like complex of bounded edges and vertices; for $P$35, it is a $P$36-dimensional cell complex embedded in a higher-dimensional unbounded polyhedron (Eppstein et al., 2011).

This perspective refines classical face-count bounds for polytopes. The Upper Bound Theorem controls the worst-case complexity of a $P$37-dimensional polytope by quantities of order $P$38, which grow rapidly with $P$39. Here, by contrast, the complexity of the bounded part is governed by $P$40, the maximum bounded-face dimension. The data explicitly frames this as an instance-based complexity parameter: the instance is easy if its bounded subcomplex is low-dimensional, even when $P$41 is large.

The paper also records projective and dual interpretations of the bounded subcomplex of a halfspace intersection. Via projective transformations and duality, it is equivalent to the faces of a polytope lying above a linear threshold, the faces disjoint from a given facet, or the faces nonadjacent to a specified vertex in the dual. This connects the bounded intersection viewpoint to thresholded face sets in convex polytopes and thereby to applications including Delaunay triangulations, tight spans of metrics, and combinatorial optimization polytopes.

A further implication is algorithmic rather than purely enumerative. Because each bounded vertex is obtained as an $P$42-minimum on an affine subspace defined by at most $P$43 supporting hyperplanes, the bounded geometry behaves combinatorially like that of a $P$44-dimensional polytope with $P$45 vertices, even though the ambient polyhedron lives in $P$46.

6. Terminological scope and other meanings of “intersection property”

The designation “bounded intersection property” is not standard across mathematics, and the paper on halfspace intersections does not itself use that phrase. In the present convex-geometric usage, it is best understood as an editorial consolidation for the condition that all bounded faces of a halfspace intersection have dimension at most $P$47 (Eppstein et al., 2011).

The term “intersection property” is used in several unrelated senses elsewhere on arXiv. In interaction decomposition over posets, it denotes an inclusion

$P$48

for sums of subspaces over lower sets, characterizing decomposability of the functor $P$49 (Sergeant-Perthuis, 2019). In local biholomorphism groups, the “uniform intersection property” means that local intersection multiplicities $P$50 take only finitely many values under the group action, and in dimension $P$51 this is equivalent to finite determination (Ribón, 2018). In Banach space geometry, Mazur-type intersection properties describe when bounded convex sets are intersections of closed balls or of closed convex hulls of finitely many balls, together with strong and uniform variants relative to a compatible class of bounded sets (Bandyopadhyay et al., 2023, Basu et al., 10 Mar 2025). In mixed-integer optimization, an intersection cut from a bounded lattice-free polytope has finite split rank exactly when the integer points on its boundary satisfy the “2-hyperplane property” (Basu et al., 2017). In tensor-network theory, the intersection property for matrix product states ensures that the parent Hamiltonian ground space is an MPS space with degeneracy bounded by $P$52 (Garre-Rubio et al., 28 Jan 2025).

This terminological spread matters because it separates several distinct ideas: bounded-face dimension in convex polyhedra, sum-intersection conditions in subspace lattices, uniform bounds on local intersection multiplicities, ball-intersection representation in Banach spaces, split-rank finiteness for intersection cuts, and MPS ground-space structure. The convex-geometric bounded intersection property is therefore specific to the theory of high-dimensional halfspace intersections with low-dimensional bounded subcomplexes.