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Bartłomiej Zawalski

Publications and source records attributed to Bartłomiej Zawalski.

8 recordsLinked to original sources
arXiv2026

Integral inequalities for $α$-convolutions of $α$-concave functions

Classical sumset inequalities originating in additive combinatorics admit geometric analogues for convex bodies in finite-dimensional real vector spaces, as recently developed by Fradelizi and two of the authors. We develop integral analogues for geometric $α$-concave functions under $α$-convolutions, an operation that arises naturally in the ``geometrization of probability'' program. In particular, we establish a Plünnecke--Ruzsa-type inequality, as well as sharp analogues of sum-difference and Ruzsa triangle inequalities, for $α$-convolutions of $α$-concave functions. We also prove a sharp Rogers--Shephard-type inequality and characterize its equality cases for $α$-concave functions, bridging the log-concave case studied by Alonso-Gutiérrez, González-Merino, Jiménez, and Villa and the quasi-concave case studied by Colesanti.

math.FA↗
arXiv2026

The $H^1(\T^d)$-multiplier norm of Rajchman measures, Zafran-type phenomena, and Walsh--Stieltjes series

We study convolution by complex Rajchman measures as Fourier multipliers on the mean-zero part of the classical isotropic Hardy space $H^1(\T^d)$. Our main technical result gives a lower bound for the multiplier norm in terms of the amount of mass carried by sets of small lower Minkowski dimension. The estimate holds for every $d\ge1$; the argument is first established for real measures and then extended to complex measures by a finite phase decomposition. Since the testing functions have mean zero, a corresponding lower bound also holds on the full isotropic Hardy space. The estimate yields multidimensional Zafran-type criteria for non-natural spectra and, in particular, shows that positive Rajchman probability measures carried by sets of lower Minkowski dimension zero induce convolution operators with non-natural spectrum. In dimension one, for real measures, we also obtain the analogous estimate on the analytic space $H^1_0(\T)$ and frequency-localized failures of the natural-spectrum property. Finally, the binary form of the underlying combinatorial lemma gives Haar block estimates, which transfer to uncertainty principles and carrier-dimension results for Walsh--Stieltjes coefficients.

math.FA↗
arXiv2026

On flat shadow boundaries from point light sources and the characterization of ellipsoids

In his classical work, W. Blaschke proved that a convex body whose shadow boundaries are flat for every direction of parallel illumination must be an ellipsoid. An extension recently proposed by I. Gonzalez-García, J. Jerónimo-Castro, E. Morales-Amaya, and D.J. Verdusco-Hernández predicts that the same conclusion holds for illumination by point light sources located on a hypersurface enclosing the body. We confirm this conjecture for convex bodies with sufficiently smooth boundaries. We further develop a duality framework relating illumination by point light sources to classical symmetry properties of hyperplane sections, linking several known and conjectured characterizations of quadrics from these complementary viewpoints.

math.MG↗
arXiv2026

Differential characterization of quadratic surfaces

Let $f\in W^{3,1}_{\mathrm{loc}}(Ω)$ be a function defined on a connected open subset $Ω\subseteq\mathbb R^2$. We will show that its graph is contained in a quadratic surface if and only if $f$ is a weak solution to a certain system of third-order partial differential equations unless the Hessian determinant of $f$ is non-positive everywhere on $Ω$. Moreover, we will prove that the system is, in a sense, the simplest possible in a wide class of differential equations, which will lead to the classification of all polynomial partial differential equations satisfied by parametrizations of generic quadratic surfaces. Although we will mainly use the tools of linear and commutative algebra, the theorem itself is also somewhat related to holomorphic functions.

math.AP↗
arXiv2025

On the homothety conjecture for the body of flotation and the body of buoyancy on a plane

We investigate several closely related "homothety conjectures" for convex bodies on a plane. Using the modern language of differential geometry, we systematically derive the fundamental properties of bodies of flotation, bodies of buoyancy, and bodies of illumination. As a direct consequence of our results, we show that if the body of flotation is homothetic to the body of buoyancy, and if every chord of flotation cuts off from the boundary exactly $\frac{1}{3}$ of its total affine arc length, then $K$ is an ellipse. We also provide natural affine counterparts of the classical theorems on the floating body problem from the Scottish Book due to H. Auerbach. In particular, we obtain an affine counterpart of Zindler carousels introduced by J. Bracho, L. Montejano, and D. Oliveros.

math.DG↗
arXiv2024

On star-convex bodies with rotationally invariant sections

We will prove that an origin-symmetric star-convex body $K$ with sufficiently smooth boundary and such that every hyperplane section of $K$ passing through the origin is a body of affine revolution, is itself a body of affine revolution. This will give a positive answer to the recent question asked by G. Bor, L. Hernández-Lamoneda, V. Jiménez de Santiago, and L. Montejano-Peimbert, though with slightly different prerequisites.

math.MG↗
arXiv2023

On separably integrable symmetric convex bodies

An infinitely smooth symmetric convex body $K\subset\mathbb R^d$ is called $k$-separably integrable, $1\leq k<d$, if its $k$-dimensional isotropic volume function $V_{K,H}(t)=\mathcal H^d(\{\boldsymbol x\in K:\mathrm{dist}(\boldsymbol x,H^\perp)\leq t\})$ can be written as a finite sum of products in which the dependence on $H\in\mathrm{Gr}(k,\mathbb R^d)$ and $t\in\mathbb R$ is separated. In this paper, we will obtain a complete classification of such bodies. Namely, we will prove that if $d-k$ is even, then $K$ is an ellipsoid, and if $d-k$ is odd, then $K$ is a Euclidean ball. This generalizes the recent classification of polynomially integrable convex bodies in the symmetric case.

math.MG↗