Symmetrizations of Ball-Bodies

TL;DR

This paper investigates symmetrization procedures for ball-bodies, revealing convexity properties, effects of Steiner symmetrization on dual volume, and providing counterexamples in higher dimensions.

Contribution

It introduces new insights into symmetrization effects on ball-bodies, including convexity results in 2D and counterexamples in 3D showing limitations of Steiner symmetrization.

Findings

Steiner symmetrization increases dual volume.

In 2D, Steiner symmetrals of ball-bodies remain ball-bodies.

Counterexamples in 3D show Steiner symmetrization can produce sets outside the class.

Abstract

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.