Systolic Ratio of Polytopes

• The systolic ratio of polytopes is a quantitative invariant defined by comparing the squared minimal cycle length or Ekeland–Hofer–Zehnder capacity to the polytope's volume.
• It links symplectic geometry with combinatorial optimization through explicit formulas involving facet normals and decomposition lengths.
• Bounds related to Viterbo’s conjecture and rigidity results provide practical insights into geometric topology, computational geometry, and metric regularity.

The systolic ratio of polytopes is a central quantitative invariant in both geometric topology and symplectic geometry. It measures the efficiency of a polytope with respect to its metric or symplectic properties, analogous to classical systolic invariants for smooth manifolds. Recent research has illuminated deep connections between the systolic ratio, symplectic capacities (including the Ekeland–Hofer–Zehnder capacity), combinatorial geometry, and optimization, while addressing questions arising from Viterbo’s conjecture and establishing explicit bounds for key classes of polytopes.

1. Definitions and Historical Development

The systolic ratio in the context of polytopes arises primarily in two settings:

• Metric Systolic Ratio: For polyhedral surfaces, this is the ratio of the squared length of the shortest non-contractible cycle (analogous to the systole on Riemannian surfaces) to the area or volume.
• Symplectic Systolic Ratio: For convex polytopes K ⊂ ℝ²ⁿ, defined via the Ekeland–Hofer–Zehnder capacity as

$$\operatorname{sys}(K) = \frac{(c_{\mathrm{EHZ}}(K))^{n}}{n! \cdot \mathrm{Vol}(K)}$$

where $c_{\mathrm{EHZ}}(K)$ is the minimal action of a closed characteristic on ∂K.

Classically, systolic ratios were introduced in Riemannian settings by Loewner and Pu, with significant generalizations in combinatorial and symplectic contexts for polytopes. The notion has evolved to capture the interplay between combinatorial complexity and geometric invariants.

2. Symplectic Capacities and the Systolic Ratio

In symplectic geometry, symplectic capacities are invariants used to obstruct embeddings and quantify symplectic size. The Ekeland–Hofer–Zehnder (EHZ) capacity is pivotal for polytopes:

• For a convex polytope K, $c_{\mathrm{EHZ}}(K)$ is defined via the minimal action of closed characteristics on the boundary $\partial K$.
• The paper "A Bound on the Symplectic Systolic Ratio of Polytopes in Even-Dimensional Euclidean Space" (Zediker, 23 Sep 2025) establishes combinatorial formulas for $c_{\mathrm{EHZ}}(K)$, connecting the symplectic properties directly to geometric data: matrix of face normals $N$, symplectic form $\omega(\cdot,\cdot)$, and combinatorics ($L(K,s)$, the number of required cuts to decompose K into simplices).

For simplices $S\subset\mathbb{R}^{2n}$, the systolic ratio is given sharply by

$\operatorname{sys}(S) \leq \frac{(2n)!}{n!(2n)^n}$

attaining equality for the standard simplex. More general polytopes admit bounds parameterized by the decomposition length $L(K,s)$.

3. Bounds and Extremality: Viterbo's Conjecture and Beyond

Viterbo’s Conjecture posited that $\operatorname{sys}(K)\leq 1$ for every convex domain K, with equality for the ball. This has been disproven in full generality by Haim–Kislev (see [HKCOUNTEREXAMPLE]), although tighter universal bounds remain an open area:

• The sharp bound for simplices is dimension-dependent and tends to zero as $n\to\infty$.
• For an arbitrary polytope K (with decomposition length $L$),

$$\operatorname{sys}(K) \leq \frac{(2n)!}{(2n)^n n!}(L+1)^{n-1}$$

This quantifies the trade-off between combinatorial complexity and attainable systolic ratio.

The failure of Viterbo’s conjecture suggests that higher systolic ratios are achievable for certain polytopes, specifically those that possess geometric or combinatorial irregularities not present in the ball.

4. Combinatorial and Algebraic Perspective

The combinatorial nature of polytopes plays a crucial role in systolic bounds:

• The matrix of facet normals A for a $d$-dimensional polytope is a central object; the smallest singular value $\sigma_{\min}(A)$ majorizes the degeneracy ratio (a metric analog of systolic optimality), see (Cvetković et al., 2019). While not topologically defined, this ratio (inradius/diameter) is a strong indicator of geometric regularity.
• The partitioning of K into simplices via hyperplane cuts (and the associated length $L(K,s)$) directly affects the upper bound of the systolic ratio—more cuts correspond to potentially higher combinatorial inefficiency, reducing the systolic efficiency.

For algorithmic computation (e.g., translation surfaces and origamis—see (Columbus et al., 2018)), combinatorial graphs encoding geodesic connections and saddle connections play an analogous role in finding systolic cycles in discrete settings.

5. Systolic Inequalities and Rigidity

Systolic inequalities for polytopes share structural similarities with classical results in Riemannian geometry (e.g., Pu’s inequality for RP² (Katz et al., 2020), sharp bounds for surfaces of given genus (Akrout et al., 2013), and spindle orbifolds (Lange et al., 2021)). In each case:

• The systolic ratio is maximized by highly symmetric (often "Zoll" or "Besse") metrics or configurations.
• Rigidity results are possible when defect or remainder terms vanish (as in the variance formulation for the projective plane).
• For polytopes, achieving the upper bound typically requires geometric regularity and symmetry similar to the simplex or the ball.

The existence of remainder terms or stability-type inequalities suggests that metrics or polyhedral geometries nearly attaining the bound are close (in the appropriate geometric topology) to the extremal configuration.

6. Implications and Applications

The systolic ratio has significant implications across disciplines:

• In symplectic topology, universal inequalities constrain embeddings and capacities.
• In computational geometry, these invariants guide mesh regularity and stability in finite element analysis (Cvetković et al., 2019).
• In combinatorial optimization, understanding the relationship between the geometry and the combinatorial structure (partitioning, normal matrices) informs algorithmic approaches to polytope partitioning and decomposition.

For higher-dimensional polytopes, ascertaining the precise systolic bound remains computationally expensive (the computation of $c_{\mathrm{EHZ}}(K)$ is NP-hard), suggesting avenues for approximation and combinatorial estimation.

7. Open Problems and Future Directions

Open areas for research highlighted in recent work include:

• Whether a universal constant $\alpha > 0$ exists for all polytopes with unit volume, independent of dimension and decomposition complexity (Zediker, 23 Sep 2025).
• The development of efficient algorithms for approximating symplectic capacities in high dimensions, in light of the computational barrier [Leipold & Vallentin].
• The investigation of rigidity phenomena for polytopal systolic ratios; specifically, characterizing when attaining or nearly attaining the bound imposes strong structural constraints on the polytope.
• Extending the systolic ratio paradigm to more general piecewise-flat or non-polyhedral domains and relating it to metric regularity and topological invariants.

In conclusion, the systolic ratio of polytopes encapsulates deep geometric and combinatorial properties, linking symplectic invariants, metric geometry, and algorithmic structure. The field continues to be shaped by the resolution of conjectures, explicit dimensional bounds, and the interplay of symmetry, combinatorial complexity, and geometric efficiency.