Core Concepts

A polyhedron is HV-symmetric if and only if it contains the origin, meaning it is bipolar; this property has implications for vertex and facet enumeration problems.

Abstract

Avis, D. (2024). HV-symmetric polyhedra and bipolarity. *arXiv preprint arXiv:2406.03698v2*.

This paper explores the concept of HV-symmetry in polyhedra and its relationship to the well-established concept of bipolarity. The author aims to demonstrate that a polyhedron is HV-symmetric if and only if it contains the origin, making it bipolar.

The author utilizes mathematical proofs and definitions to establish the equivalence between HV-symmetry and bipolarity in polyhedra. The paper relies on concepts like H-representation, V-representation, polarity of convex sets, and the bipolar equation. Examples are provided to illustrate the definitions and support the arguments.

The paper proves that a polyhedron is HV-symmetric if and only if it contains the origin, meaning it is also bipolar. This finding is significant because it simplifies the determination of HV-symmetry by eliminating the need to compute both V(P) and V(Q) as defined in the paper.

The equivalence between HV-symmetry and bipolarity in polyhedra has implications for computational geometry problems like vertex and facet enumeration. Specifically, when a polyhedron contains the origin, the lifting process typically used in converting between H-representation and V-representation is unnecessary, potentially leading to faster computation times for these problems.

This research contributes to a deeper understanding of the properties of polyhedra and their representations. The established link between HV-symmetry and bipolarity simplifies the identification of HV-symmetric polyhedra and offers potential computational advantages in solving geometric problems related to these objects.

While the paper proves the equivalence of HV-symmetry and bipolarity, it raises questions about the efficiency of applying this knowledge in algorithms for vertex and facet enumeration. Further research could investigate whether utilizing the unlifted representation for bipolar polyhedra consistently leads to faster computation times compared to the traditional lifted approach.

To Another Language

from source content

arxiv.org

Stats

Quotes

"It is well known and often stated that polytopes that contain the origin in their interior and pointed polyhedral cones are HV -symmetric. It seems to be less well known that, more generally, a polyhedron is HV -symmetric if and only if it contains the origin, in other words it is bipolar."

Key Insights Distilled From

by David Avis at **arxiv.org** 10-10-2024

Deeper Inquiries

Answer:
The insights into HV-symmetry offered by this paper can potentially benefit various areas within computational geometry and computer graphics:
Collision Detection: Determining if two objects intersect is a fundamental problem in collision detection. For convex objects, algorithms often leverage the objects' H-representations or V-representations. Knowing that a polyhedron is bipolar (and thus HV-symmetric) allows for choosing the most efficient representation for the specific algorithm, potentially reducing computational overhead.
Convex Hull Computations: Algorithms for computing the convex hull of a point set can be optimized by exploiting HV-symmetry. For instance, if the input points are known to form a bipolar polyhedron, the algorithm can directly compute either the H-representation or V-representation, as the other can be easily derived.
Mesh Simplification: In computer graphics, simplifying complex meshes while preserving their overall shape is crucial for efficient rendering. Understanding the HV-symmetry of a polyhedron can guide the simplification process. For example, if a mesh represents a bipolar polyhedron, simplification algorithms can prioritize preserving vertices and facets that contribute significantly to both representations.
Shape Analysis: HV-symmetry can be a valuable tool for analyzing and classifying 3D shapes. The presence or absence of this symmetry, along with properties of the polar polyhedron, can provide insights into the object's structure and potential symmetries. This information can be used in shape recognition, retrieval, and classification tasks.

Answer:
While the paper suggests that using the unlifted representation for bipolar polyhedra in vertex and facet enumeration might seem computationally advantageous, there are situations where it could lead to decreased efficiency:
Degeneracy: The performance of pivot-based algorithms like reverse search is significantly affected by the presence of degeneracies in the polyhedron. Degeneracies can lead to an exponential increase in the number of bases visited by the algorithm. It's possible that the unlifted representation of a bipolar polyhedron might introduce or exacerbate degeneracies, leading to worse performance compared to the lifted representation.
Implementation Details: The efficiency gains from using the unlifted representation depend on the specific implementation of the vertex and facet enumeration algorithm. Some implementations might be optimized for handling the lifted representation, and using the unlifted representation might require modifications that could negatively impact performance.
Problem-Specific Structure: The structure of the specific bipolar polyhedron being analyzed plays a crucial role. In some cases, the lifted representation might implicitly exploit certain structural properties of the polyhedron that are not readily apparent in the unlifted representation, leading to faster computation.
Therefore, while the theoretical possibility of improved efficiency using the unlifted representation for bipolar polyhedra exists, a careful empirical analysis considering the factors mentioned above is crucial to determine the optimal approach for a given problem instance.

Answer:
The presence or absence of the origin significantly influences a polyhedron's properties and computational complexity, particularly concerning HV-symmetry and bipolarity. Several other fundamental geometric concepts similarly impact our understanding of these objects:
Convexity: The property of convexity is paramount in the study of polyhedra. Many efficient algorithms and elegant theorems are only applicable to convex polyhedra. Non-convex polyhedra often require more complex and computationally intensive techniques.
Dimensionality: The dimension of the space in which a polyhedron resides plays a crucial role. Algorithms and their complexity often depend on the dimension. For instance, vertex enumeration for a polytope (bounded polyhedron) is polynomial-time solvable in 2D and 3D but becomes NP-hard in higher dimensions.
Combinatorial Structure: The arrangement of vertices, edges, and facets, captured by the face lattice of a polyhedron, significantly influences its properties. Properties like simplicity (each vertex belonging to exactly d facets in d-dimensional space) or simpliciality (each facet being a simplex) can simplify algorithms and lead to stronger theoretical results.
Symmetry Groups: The presence of symmetries, described by the polyhedron's symmetry group, can be exploited for efficient representation, analysis, and manipulation. For example, regular polyhedra, possessing a high degree of symmetry, have well-defined properties and are computationally easier to handle.
Degeneracies: The presence of degeneracies, such as multiple vertices lying on the same hyperplane or facets with more than d vertices in d-dimensional space, can significantly impact the complexity of algorithms. Handling degeneracies often requires careful consideration and can increase the computational cost.
Understanding these fundamental geometric concepts is crucial for developing efficient algorithms, proving theoretical results, and gaining deeper insights into the properties and behavior of polyhedra in various computational geometry and computer graphics applications.

0