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Adam Białożyt

Publications and source records attributed to Adam Białożyt.

10 recordsLinked to original sources
arXiv2026

Directional curvature and medial axis

The medial axis $M_X$ of a closed set $X\subset \mathbb{R}^n$ is the set of points from the ambient space that admit more than one closest point in $X$. We study the problem of reaching the singularities, i.e. of characterising the points of the set $\overline{M_X}\cap X$. In order to tame the geometry, we assume that $X$ is definable in a polynomially bounded structure and obtain a general criterion based on a generalisation of the notion of superquadraticity previously introduced by Birbrair and Denkowski for $C^1$-smooth hypersurfaces and extended to any codimension by Białożyt. We do not require any smoothness as we achieve our goal by introducing a notion of directional curvature in some naturally chosen camber directions. This allows us in particular to complete the study of the plane case.

math.MG↗
arXiv2026

Sets Reconstructable with Medial Axis

The medial axis of a closed set is well established tool in pattern recognition, cherished for its power of reconstruction of domains. In this article we fill this gap answering the question which sets precisely are reconstructible from the medial axis information. We do not assume any additional structure of considered sets besides them being closed in n-dimensional Euclidean space.

math.MG↗
arXiv2025

Medial Axis in Pseudo-Euclidean Spaces

We investigate the notion of the medial axis for pseudo-Euclidean spaces. For most of the article, we follow the path of Birbrair and Denkowski's article "Medial Axis and Singularities", checking its feasibility in the new context.

math.DG↗
arXiv2025

Algebraic normalisation

We present a strictly geometric c-algebraic version of the analytic set normalisation. With the introduced tool we prove the Nullstellensatz for c-algebraic functions and study the growth exponent of a c-algebraic function.

math.AG↗
arXiv2022

On the singular points approached by the medial axis

This paper develops the notion of superquadracity defined by L.Birbrair and M.Denkowski for subsets of R^n. In this regard, the main theorem of the paper establishes the relation between the superquadracity and non-empty intersection of the set and the closure of its medial axis. The further investigation concerns non C1 smooth points of the set.

math.MG↗
arXiv2022

The Kuratowski convergence of medial axes and conflict sets

This paper consists of two parts. In the first one we study the behaviour of medial axes (skeletons) of closed sets in a connected complete Riemannian manifold $\mathcal{M}$ under deformations. The second one is devoted to a similar study of conflict sets. We apply a new approach to the deformation process. Instead of seeing it as a `jump' from the initial to the final state, we perceive it as a continuous process, expressed using the Kuratowski convergence of sets (hence, unlike other authors, we do not require any regularity of the deformation). Our main `medial axis inner semi-continuity' result has already proved useful, as it was used to compute the tangent cone of the medial axis with application in singularity theory.

math.MG↗
arXiv2021

The tangent cone, the dimension and the frontier of a medial axis

This paper aims to establish a relation between the tangent cone of the medial axis of X at a given point a of R^n$ and the medial axis of the set of points in X realising the distance d(a,X). As a consequence, a lower bound for the dimension of the medial axis of X in terms of the dimension of the medial axis of m(a) is obtained. This appears to be the missing link to the full description of the medial axis' dimension. Further study of potentially troublesome points on the frontier of the medial axis is also provided, resulting in their characterisation in terms of the reaching radius.

math.MG↗
arXiv2020

On the growth exponent of c-holomorphic functions with algebraic graphs

This paper is the first of a series dealing with c-holomorphic functions defined on algebraic sets and having algebraic graphs. These functions may be seen as the complex counterpart of the recently introduced \textit{regulous} functions. Herein we study their growth exponent at infinity. A general result on injectivity on fibres of an analytic set together with a theorem of Tworzewski and Winiarski gives a bound for the growth exponent of a c-holomorphic function with algebraic graph in terms of the projective degrees of the sets involved. We prove also that algebricity of the graph is equivalent to the function being the restriction of a rational function (a Serre-type theorem). Then we turn to considering generically finite c-holomorphic mappings with algebraic graphs and we prove a Bézout-type theorem. We also study a particular case of the Łojasiewicz inequality at infinity in this setting.

math.CV↗