arXiv subjects

Michał Zwierzyński

Publications and source records attributed to Michał Zwierzyński.

12 recordsLinked to original sources

arXiv2026

Shuffle Squares in Differentiable Words

Experiments on binary run-length differentiability lead to sharp computer-assisted criteria for shuffle squares. Every $C^3$-word of length greater than $16$ is a shuffle square exactly when both letter multiplicities are even. All $34$ nonempty even-Parikh exceptions are smooth and persist in every higher differentiability class. For $C^2$ the sharp threshold is $48$, with $212$ nonempty exceptions. At level $C^1$ no global parity threshold exists, but every even-Parikh non-shuffle-square of length at least $36$ has proper nonempty shuffle-square prefixes and suffixes. Exactly $230$ nonempty even-Parikh $C^1$-words have no nonempty shuffle-square prefix. The full tree avoiding such prefixes eventually consists of $422$ periodic rays. Consequently, a nonempty Kolakoski prefix is a shuffle square exactly when both multiplicities are even, apart from lengths $4$ and $8$. We also characterize classes of morphisms reflecting shuffle squares. For doubly binary words, we determine the exact deletion distance and largest twins, and prove sharp bounds for single local repairs. Exact recurrences, residual-state checks, and separate Python programs make the finite computations reproducible.

math.CO↗

arXiv2026

Singularities of the Wigner Caustic and the Centre Symmetry Set of Frontal Curves in the Euclidean Plane

Motivated by the role of the Wigner caustic in semiclassical phase-space analysis, we extend it, together with the centre symmetry set, from regular planar curves to cooriented frontals. For an angularly regular parallel pair, the signed speed of the Wigner caustic is one half of the difference of the extended signed radii of curvature, while singular points of the finite centre symmetry set are the critical points of their projective ratio. These formulas yield criteria and explicit invariants for ordinary and higher cusps, an order-lowering relation between the two constructions, and a multiplicity-weighted extension of the classical cusp-count inequality for strictly convex ovals. We also analyse parallel pairs containing a singular frontal which is not a front. A $5/2$-cusp produces either a $5/2$- or, at a signed-radius resonance, a $5/3$-cusp on the Wigner caustic. The opposite resonance sends the centre symmetry set to infinity. We obtain the corresponding transfer results for a $5/3$-cusp and give a projective completion which resolves vanishing denominators and simultaneous finite-order zeros of the two signed radii.

math.DG↗

arXiv2026

Invariants of an Affine Reflection Transform

We introduce the affine reflection transform (ART) of an oval $C$ relative to an interior point $p$ by applying the parallel-tangent involution to chords through $p$ and taking the envelope. A dual-projective approach gives a coordinate-free singularity criterion and proves that every generic non-degenerate ART has an odd number of ordinary cusps, at least three. Its relation to the centre symmetry set and the Wigner caustic yields further cusp bounds. We define a discrete ART for convex polygons with parallel opposite sides. Its combinatorial cells admit explicit rational parametrisations. Non-degenerate cells are arcs of ellipses or hyperbolas, and never parabolas. Under refining tangent approximations, the discrete transform converges in the Hausdorff metric to the smooth one. The oriented area of the smooth and polygonal transforms is non-positive and admits a Sobolev-type interpretation. Maximising its absolute value produces an affine-invariant asymmetry measure and a set-valued affine centre. We establish boundary degeneration and vanishing results, characterise central symmetry, and show that the maximising locus need not lie on the centre symmetry set or Wigner caustic. Finally, we prove universal upper bounds for the normalised energy, strengthen them for ovals of constant width.

math.DG↗

arXiv2026

A Sharp Surface-Area Extension of Vaaler's Theorem

Karasev proved, in dimensions two and three, a sharp surface-area counterpart of a polyhedral extension of Vaaler's theorem [8, Theorem 1.2]. We remove the dimension restriction. More precisely, if an $n$-dimensional convex polytope $P$ contains the origin in its interior and the affine hull of every nonempty proper face of codimension $k\in\{1,\ldots,n\}$ is at distance at least $\sqrt{k}$ from the origin, then $$ \mathcal{H}^{n-1}(\partial P)\geqslant n2^n. $$ Consequently, the boundary of every $n$-dimensional linear section of the cube $[-1,1]^N$ has $(n-1)$-dimensional measure at least $n2^n$. This confirms, in all dimensions, a conjecture of Grigory M. Ivanov recorded by Karasev [8, Section 1]. The proof combines the Rogers--Karasev flag decomposition with a dimension-free Gaussian comparison for orthoschemes. Its main step is an ordered-square substitution which converts all face-distance assumptions into a pointwise domination of a single positive integral. We also disprove the naive extension to all skeletal measures, formulate a ridge-skeleton conjecture, and establish two partial results toward it.

math.MG↗

arXiv2026

Area-Normalized Pentagram Map Dynamics: Spectral Flattening and Elliptic Asymptotics

The pentagram map sends a polygon to the intersections of consecutive short diagonals. We study its shape dynamics after translating and positively rescaling each iterate to restore unsigned area and barycenter. For polygons whose dynamics is generated, after passing to a finite iterate and cyclic relabelling, by one projectivity, we prove a spectral dichotomy. A dominant real projective line yields flattening and unbounded diameter, whereas a dominant real eigenvalue with a subdominant non-real pair produces asymptotic motion on concentric homothetic ellipses. We apply this framework to pentagons and hexagons via Glick's operator and to Poncelet polygons via the Darboux--Schwartz projectivity. We obtain spectral diagrams and a Jacobi-function formula for the Poncelet return spectrum. Consequently, we recover Schwartz's long-and-thin result for strictly convex non-projectively-regular pentagons, prove a line-or-ellipse dichotomy for convex hexagons under explicit nondegeneracy assumptions, and establish flattening for strictly convex non-projectively-regular Poncelet polygons.

math.DS↗

arXiv2026

Mixed Volumes, the Wigner Caustic, and Isoperimetric Inequalities

Let $K\subset\mathbb{R}^n$ be a convex body and $C=\frac12(K+(-K))$ its central symmetral. We study the defects $\mathcal A_k(K)=W_{n-k}(C)-W_{n-k}(K)$ through the even--odd decomposition of the support function. We derive exact even mixed-volume expansions, identify their Krawtchouk transform with the mixed difference-body coefficients, and obtain a Kubota formula expressing each defect as the averaged symmetrization gain of projections. These identities yield strengthened isoperimetric inequalities. The quadratic defect is the average of the absolute oriented areas of projected Wigner caustics and obeys a sharp spherical-harmonic stability estimate. We give explicit formulas in dimensions $2$, $3$, and $4$. The four-dimensional expansion includes a sign-indefinite quartic Wigner volume; on a plane of cubic harmonics we find nonspherical constant-width bodies for which it vanishes. Finally, we prove a projection-transfer bound and an arbitrary-dimensional expansion for constant-width bodies.

math.DG↗

arXiv2026

Convex Polygons with Parallel Opposite Sides: Convergence, Reconstruction, and Isoperimetric Inequalities

In this paper, we study affine $λ$-equidistants of convex polygons with parallel opposite sides (CPPOSes), introduced by M. Craizer et al. (Polygons with Parallel Opposite Sides, Discrete & Computational Geometry 50 (2013), 474-490), and relate them to affine $λ$-equidistants of smooth convex curves. We show how, and under which conditions, the original CPPOS can be reconstructed from its Wigner caustic and centre symmetry set. We prove a discrete version of the improved isoperimetric inequality: $$L(\mathcal{P})^2\geqslant 8n\tan\left(\fracπ{2n}\right)\cdot\big( A(\mathcal P)+2\left|A^\ast\left(\mathrm{E}_{0.5}(\mathcal P)\right)\right|\big),$$ where $\mathcal{P}$ is a CPPOS with $2n$ vertices, $L(\mathcal{P})$ is its perimeter, $A(\mathcal{P})$ is the area enclosed by $\mathcal{P}$, and $A^\ast\left(\mathrm{E}_{0.5}(\mathcal P)\right)$ denotes the oriented area of the Wigner caustic of $\mathcal{P}$. Moreover, equality holds if and only if $\mathcal{P}$ is an equiangular CPPOS of constant width. We also prove sharp area estimates for the oriented areas of the Wigner caustic and the centre symmetry set of a CPPOS. More precisely, we show that the absolute value of the oriented area of the Wigner caustic is at most one quarter of the area of $\mathcal{P}$, while the absolute value of the oriented area of the centre symmetry set is at most the area of $\mathcal{P}$. Both bounds are sharp.

math.DG↗

arXiv2026

The $k$th Order Preserving Sets and Isoperimetric Type Inequalities for Planar Ovals

In this work, we introduce and investigate a new class of sets, the \textit{$k$th Order Preserving Sets}, arising naturally from the Fourier analysis of support functions associated with hedgehogs. Specifically, we focus on sets whose support functions possess a Fourier series that preserves only terms with positive indices divisible by a fixed $k$. We explore the geometry of the \textit{$k$th Order Midpoint Set}, defined as the set of centroids of all equiangular $k$-gons circumscribed about a given hedgehog. This set captures essential structural and symmetry-related features of the underlying geometric configuration. We study the geometric properties of such sets and, in particular, establish an isoperimetric-type inequality relating the perimeter and area of a region bounded by a simple smooth convex closed curve (an oval) $\mathcal{O}$: \[ L_{\mathcal{O}}^2 - 4πA_{\mathcal{O}} \geqslant 4π|A_{\mathcal{P}_k}| + 2π|A_{Ω_{\mathcal{O},k}}|, \] where $L_{\mathcal{O}}$ denotes the length (perimeter) of $\mathcal{O}$, $A_{\mathcal{O}}$ is the area of the region enclosed by $\mathcal{O}$, $A_{\mathcal{P}_k}$ is the oriented area of the associated $k$th Order Preserving Set $\mathcal{P}_k$, and $A_{Ω_{\mathcal{O},k}}$ is the oriented area of the associated $k$th Order Midpoint Set $Ω_{\mathcal{O},k}$. Moreover, we characterize the equality case: the inequality becomes an equality if and only if every equiangular circumscribed $k$-gon around $\mathcal{O}$ is a~regular $k$-gon with its center of mass located at the Steiner point of $\mathcal{O}$.

math.DG↗

arXiv2024

The Geometry of the Centre Symmetry Set of a Planar Curve

The Centre Symmetry Set of a planar curve $M$ is the envelope of affine chords of $M$, i.e. the lines joining points on $M$ with parallel tangent lines. In this paper we study global geometrical properties of this set including the number of singularities and the number of asymptotes.

math.DG↗

arXiv2024

Artistic Aspects of the Wigner Caustic and the Centre Symmetry Set

The Wigner caustic and the Centre Symmetry Set of a closed smooth planar curve are known singular sets which generically admit only cusp singularities. Applications of these objects in semi-classical quantum physics, in chaos theory, in singularity theory, in convex geometry, have been studied since the 1970s until today. These sets can be viewed as envelopes of special families of lines and thanks to that they have many geometric artistic values.

math.GM↗

arXiv2018

The Gauss-Bonnet Theorem for coherent tangent bundles over surfaces with boundary and its applications

In [31,32,33] the Gauss-Bonnet formulas for coherent tangent bundles over compact oriented surfaces (without boundary) were proved. We establish the Gauss-Bonnet theorem for coherent tangent bundles over compact oriented surfaces with boundary. We apply this theorem to investigate global properties of maps between surfaces with boundary. As a corollary of our results we obtain Fukuda-Ishikawa's theorem. We also study geometry of the affine extended wave fronts for planar closed non singular hedgehogs (rosettes). In particular, we find a link between the total geodesic curvature on the boundary and the total singular curvature of the affine extended wave front, which leads to a relation of integrals of functions of the width of a resette.

math.DG↗

arXiv2016

The improved isoperimetric inequality and the Wigner caustic of planar ovals

The classical isoperimetric inequality in the Euclidean plane $\mathbb{R}^2$ states that for a simple closed curve $M$ of the length $L_{M}$, enclosing a region of the area $A_{M}$, one gets \begin{align*} L_{M}^2\geqslant 4πA_{M}. \end{align*} In this paper we present the improved isoperimetric inequality, which states that if $M$ is a closed regular simple convex curve, then \begin{align*} L_{M}^2\geqslant 4πA_{M}+8π\left|\widetilde{A}_{E_{\frac{1}{2}}(M)}\right|, \end{align*} where $\widetilde{A}_{E_{\frac{1}{2}}(M)}$ is an oriented area of the Wigner caustic of $M$, and the equality holds if and only if $M$ is a curve of constant width. Furthermore we also present a stability property of the improved isoperimetric inequality (near equality implies curve nearly of constant width). The Wigner caustic is an example of an affine $λ$-equidistant (for $\displaystyleλ=\frac{1}{2}$) and the improved isoperimetric inequality is a consequence of certain bounds of oriented areas of affine equidistants.

math.DG↗