arXiv subjects
Publications and source records attributed to Martin Winter.
We study rational generalized barycentric coordinates (rational GBCs) on polytopes. We recall their essential theory and clarify some statements from the literature. We study polytopes that admit GBCs with low degree denominators (and hence, low degree (affine) adjoint polynomials). We give a full classification of polytopes that admit polynomial barycentric coordinates (i.e. constant adjoints).
We give a new proof that 1-extension preserves generic local rigidity of bar-joint frameworks in all dimensions. The proof does not use a special position argument, which makes the approach potentially useful for X-replacement in 3-space. Instead, the balancing conditions recently introduced by the first author use directed paths in the line graph, and by modifying the end of these paths, we transfer a self-stress on the 1-extended graph back to the original. Treating the new vertex as variable allows us to extract the balancing condition of the smaller graph as one coefficient of a homogeneous quadratic. A Lean 4 formalization of the main results accompanies the paper.
Given a convex polytope $P\subset\Bbb R^d$ with $\mathcal F(P)$ many flags, we prove $$\operatorname{vol}(P) \operatorname{vol}(P-P)^\circ \le \frac{\mathcal F(P)}{(d!)^2}.$$ This implies the following relation between Mahler volume and the number of flag conjectured by Freij, Schmitt, Schymura and Ziegler: for a centrally symmetric polytope $P\subset\Bbb R^d$ holds $$\operatorname{vol}(P)\operatorname{vol}(P^\circ) \le \frac{2^d}{(d!)^2} \mathcal F(P).$$ This shows that the Mahler conjecture implies Kalai's flag conjecture.
The infinitesimal Stoker conjecture states that an infinitesimal deformation of a convex polytope that preserves all dihedral angles also preserves all normal directions up to isometry. We give a short polyhedral proof, self-contained up to assuming spectral and geometric properties of the Izmestiev matrix.
We study the depth required by ReLU networks to exactly represent piecewise linear functions, focusing specifically on the maximum function. This problem has recently received significant attention in both the ML and TCS literature. We prove that $\max_n(x)=\max\{x_1,\ldots,x_n\}$ is exactly representable with two hidden layers for every $n\leq 12$. Previously, this was only known up to $n\leq5$ [Bakaev, Brunck, Hertrich, Stade, Yehudayoff, STOC'26]. We obtain our constructions through an exact computer-assisted search within a space of candidate solutions: After a symmetry reduction, we obtain a finite system of linear equations over $\mathbb{Q}$ such that any solution yields a valid representation of the maximum function. The resulting constructions have a structured first hidden layer, which enables recursive substitution into deeper networks. This yields an exact ReLU representation of $\max_n$ with at most $\lceil \log_6(n/2) \rceil+1$ hidden layers. Consequently, every continuous piecewise-linear function on $\mathbb{R}^d$ admits an exact representation with at most $\lceil\log_6((d+1)/2)\rceil+1$ hidden layers; in particular, two hidden layers suffice for $d\leq 11$. Again, these results improve upon [Bakaev, Brunck, Hertrich, Stade, Yehudayoff, STOC'26], who proved analogous logarithmic bounds with base three.
A coned polytope framework (CPF) is the bar-joint framework obtained from the 1-skeleton of a convex polytope by coning over some interior point. It was recently shown that CPFs are rigid, though the exact order of rigidity remained open. In this paper we introduce the Wachspress stress and use it to show that CPFs are prestress stable, in particular, second-order rigid. To this end, we resolve the stress-flex conjecture in the case of the Wachspress stress by identifying its dual formulation as a corollary of a vector-valued Schläfli-type formula introduced by Schlenker and Souam. We give a new and purely discrete-geometric proof of this generalized Schläfli formula.
We study deformations of polytopes that preserve edge lengths and face coplanarities. This gives rise to a notion of rigidity, for which we develop a second-order theory together with an effective algorithm for testing second-order rigidity symbolically. The strength of these new tools is demonstrated on several polytope classes, including the previously intractable regular dodecahedron, which we find to be rigid. We also find that a single tested polytope evades our techniques and will therefore provide a simple test case for future developments of tools of even higher order. This paper also contains a study of so-called edge-length perturbations. We point out connections between rigidity and the ability to realize polytopes with slightly perturbed edge lengths. To explore these connections in practice, we dedicate a section to another case study of the regular dodecahedron. We investigate the local behavior of its realization space with a view towards edge-length perturbations, singularities and generic global rigidity.
Recent advancements in Gaussian Splatting (3DGS) have introduced various modifications to the original kernel, resulting in significant performance improvements. However, many of these kernel changes are incompatible with existing datasets optimized for the original Gaussian kernel, presenting a challenge for widespread adoption. In this work, we address this challenge by proposing an alternative kernel that maintains compatibility with existing datasets while improving computational efficiency. Specifically, we replace the original exponential kernel with a polynomial approximation combined with a ReLU function. This modification allows for more aggressive culling of Gaussians, leading to enhanced performance across different 3DGS implementations. Our results show a notable performance improvement of 4 to 15% with negligible impact on image quality. We also provide a detailed mathematical analysis of the new kernel and discuss its potential benefits for 3DGS implementations on NPU hardware.
We investigate a novel setting for polytope rigidity, where a flex must preserve edge lengths and the planarity of faces, but is allowed to change the shapes of faces. For instance, the regular cube is flexible in this notion. We present techniques for constructing flexible polytopes and find that flexibility seems to be an exceptional property. Based on this observation, we introduce a notion of generic realizations for polytopes and conjecture that convex polytopes are generically rigid in dimension $d\geq 3$. We prove this conjecture in dimension $d=3$. Motivated by our findings we also pose several questions that are intended to inspire future research into this notion of polytope rigidity.
The development of material acceleration platforms in battery research requires integrating complementary techniques and correlating heterogeneous experimental datasets. Here, this challenge is tackled in a large-scale multimodal program involving fifteen laboratories and facilities across Europe. Coordinated multi-site experiments are performed on state-of-the-art graphite / LiNiO2 Li-ion full cells to address two archetypal scientific questions: is the electrolyte composition impacting electrode properties, and how do electrode materials evolve when cells are cycled to their end-of-life? A fully standardized and centralized workflow is demonstrated, from sample production and delivery, to metadata and data handling, generating seventy-five concatenated datasets shared among all partners. Their integrated analysis shows that scientific conclusions depend critically on both the observable chosen to describe electrode properties, and the measurement technique employed. Individual experiments provide detailed information into specific aspects, such as crystal structures, redox activity, surface processes, morphology, etc., but can also function as binary diagnostic tool. Two-dimensional observable-technique patterns are introduced, in which each pixel encodes a yes, no or uncertain answer to a given scientific question. These patterns serve as multi-property metaviews, e.g. visual genotypes, enabling to classify material behavior and technique suitability according to predefined user demand and criteria, highlighting the interdependencies between measurement choices, extracted parameters and scientific interpretation. This multimodal workflow establishes a proof-of-concept for correlative analysis and underscores challenges toward fully integrated, automated and holistic approaches in energy material science.
We study the canonical form $Ω$ as a valuation in the context of scissors congruence for polytopes. We identify the degree of its numerator - the adjoint polynomial $\operatorname{adj}_P$ - as an important invariant in this context. More precisely, for a polytope $P$ we define the degree drop that measures how much smaller than expected the degree of the adjoint polynomial of $P$ is. We show that this drop behaves well under various operations, such as decompositions, restrictions to faces, projections, products and Minkowski sums. Next we define the reduced canonical form $Ω_0$ and show that it is a translation-invariant 1-homogeneous valuation on polytopes that vanishes if and only if $P$ has positive degree drop. Using it we can prove that zonotopes can be characterized as the $d$-polytopes that have maximal possible degree drop $d-1$. We obtain a decomposition formula for $Ω_0$ that expresses it as a sum of edge-local quantities of $P$. Finally, we discuss valuations $Ω_s$ that can distinguish higher values of the degree drop.
We investigate for which classes of (potentially infinite) graphs the clique dynamics is cover stable, i. e., when clique convergence/divergence is preserved under triangular covering maps. We first present an instructive counterexample: a clique convergent graph which covers a clique divergent graph and which is covered by a clique divergent graph. Based on this we then focus on local conditions (i. e., conditions on the neighbourhoods of vertices) and show that the following are sufficient to imply cover stability: local girth $\geq 7$ and local minimum degree $\geq 2$; being locally cyclic and of minimum degree $\geq 6$.
Modelling and numerical simulations play a vital role in the design and optimization of electrochemical energy storage devices. In this study, a general physics-based model is developed to describe Hybrid Asymmetric Capacitors (HACs). A one-dimensional cell is constructed with one faradaic electrode, a separator and a capacitive electrode. The model is validated using a commercial Lithium Ion Capacitor (LIC). Galvanostatic charge and discharge processes are simulated with a maximum mean relative error of 7.8 %. This suggest that this simple Ohmic model captures the key electrochemical phenomena occurring inside the LIC cell.
The clique graph $kG$ of a graph $G$ has as its vertices the cliques (maximal complete subgraphs) of $G$, two of which are adjacent in $kG$ if they have non-empty intersection in $G$. We say that $G$ is clique convergent if $k^nG\cong k^m G$ for some $n\not= m$, and that $G$ is clique divergent otherwise. We completely characterise the clique convergent graphs in the class of (not necessarily finite) locally cyclic graphs of minimum degree $δ\ge 6$, showing that for such graphs clique divergence is a global phenomenon, dependent on the existence of large substructures. More precisely, we establish that such a graph is clique divergent if and only if its universal triangular cover contains arbitrarily large members from the family of so-called "triangular-shaped graphs".
We study the potentially undecidable problem of whether a given 2-dimensional CW complex can be embedded into $\mathbb{R}^4$. We provide operations that preserve embeddability, including joining and cloning of 2-cells, as well as $Δ\mathrm Y$-transformations. We also construct a CW complex for which $\mathrm YΔ$-transformations do not preserve embeddability. We use these results to study 4-flat graphs, i.e., graphs that embed in $\mathbb{R}^4$ after attaching any number of 2-cells to their cycles; a graph class that naturally generalizes planarity and linklessness. We verify several conjectures of van der Holst; in particular, we prove that each of the 78 graphs of the Heawood family is an excluded minor for the class of 4-flat graphs.
Recently, it has been proven that a tensegrity framework that arises from coning the one-skeleton of a convex polytope is rigid. Since such frameworks are not always infinitesimally rigid, this leaves open the question as to whether they are at least prestress stable. We prove here that this holds subject to an intriguing new conjecture about coned polytope frameworks, that we call the stress-flex conjecture. Multiple numerical experiments suggest that this conjecture is true, and most surprisingly, seems to hold even beyond convexity and also for higher genus~polytopes.
Kalai's $3^d$ conjecture states that every centrally-symmetric $d$-polytope has at least $3^d$ faces. We give short proofs for two special cases: if $P$ is unconditional (that is, invariant w.r.t. reflection in any coordinate hyperplane), and more generally, if $P$ is locally anti-blocking. In both cases we show that the minimum is attained exactly for the Hanner polytopes.
We conjecture that a convex polytope is uniquely determined up to isometry by its edge-graph, edge lengths and the collection of distances of its vertices to some arbitrary interior point, across all dimensions and all combinatorial types. We conjecture even stronger that for two polytopes $P\subset\mathbb R^d$ and $Q\subset\mathbb R^e$ with the same edge-graph it is not possible that $Q$ has longer edges than $P$ while also having smaller vertex-point distances. We develop techniques to attack this question and verify it in three relevant special cases: if $P$ and $Q$ are centrally symmetric, if $Q$ is a slight perturbation of $P$, and if $P$ and $Q$ are combinatorially equivalent. In the first two cases the statements stay true if we replace $Q$ by some graph embedding $q\colon V(G_P)\to\mathbb R^e$ of the edge-graph $G_P$ of $P$, which can be interpreted as local resp. universal rigidity of certain tensegrity frameworks. We also establish that a polytope is uniquely determined up to affine equivalence by its edge-graph, edge lengths and the Wachspress coordinates of an arbitrary interior point. We close with a broad overview of related and subsequent questions.