Symmetrizations of Ball-Bodies
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| author | Artstein-Avidan, Shiri Florentin, Dan I. |
| author_facet | Artstein-Avidan, Shiri Florentin, Dan I. |
| contents | We study symmetrization procedures within the class $\mathcal S_n$ of \emph{ball-bodies}, i.e.\ intersections of unit Euclidean balls (equivalently, summands of the Euclidean unit ball, or $c$-convex sets via the $c$-duality $A\mapsto A^c$). We first examine linear parameter systems obtained by replacing the usual convex hull by the $c$-hull $A^{cc}$, deriving consequences for volume along these $c$-paths. In particular, we obtain convexity statements in special cases and in dimension $2$, and we show by example that such convexity fails in general for $n\ge 3$. We then focus on Steiner symmetrization. We prove that Steiner symmetrization increases the \emph{dual volume} and that in the planar case Steiner symmetrals of ball-bodies remain ball-bodies. In contrast, we provide an explicit example in $\RR^3$ showing that the Steiner symmetral of a ball-body need not belong to $\mathcal S_n$, and show that there are such counter-examples with arbitrarily large curvatures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14090 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Symmetrizations of Ball-Bodies Artstein-Avidan, Shiri Florentin, Dan I. Metric Geometry 52A10, 52A15, 52A20, 52A40, 51F99, 52A38, 53A04, 53A05 We study symmetrization procedures within the class $\mathcal S_n$ of \emph{ball-bodies}, i.e.\ intersections of unit Euclidean balls (equivalently, summands of the Euclidean unit ball, or $c$-convex sets via the $c$-duality $A\mapsto A^c$). We first examine linear parameter systems obtained by replacing the usual convex hull by the $c$-hull $A^{cc}$, deriving consequences for volume along these $c$-paths. In particular, we obtain convexity statements in special cases and in dimension $2$, and we show by example that such convexity fails in general for $n\ge 3$. We then focus on Steiner symmetrization. We prove that Steiner symmetrization increases the \emph{dual volume} and that in the planar case Steiner symmetrals of ball-bodies remain ball-bodies. In contrast, we provide an explicit example in $\RR^3$ showing that the Steiner symmetral of a ball-body need not belong to $\mathcal S_n$, and show that there are such counter-examples with arbitrarily large curvatures. |
| title | Symmetrizations of Ball-Bodies |
| topic | Metric Geometry 52A10, 52A15, 52A20, 52A40, 51F99, 52A38, 53A04, 53A05 |
| url | https://arxiv.org/abs/2602.14090 |