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Dan I. Florentin

Publications and source records attributed to Dan I. Florentin.

7 recordsLinked to original sources
arXiv2026

Symmetrizations of Ball-Bodies

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.

math.MG↗
arXiv2025

An in-depth study of ball-bodies

In this paper we study the class of so called `ball-bodies' in ${\mathbb R}^n$, given by intersections of translates of Euclidean unit balls (or, equivalently, summand of the Euclidean ball). We study the class along with the natural duality operator defined on it. The class is naturally linked to many interesting problems in convex geometry, including bodies of constant width and the Knesser-Poulsen conjecture. We discuss old and new inequalities of isoperimetric type and of Santaló type, in this class. We study the boundary structure of bodies in the class, Carathéodory type theorem and curvature relations. We discuss various symmetrizations with relation to this class, and make some first steps regarding problems for bodies of constant width.

math.MG↗
arXiv2024

On the Many Faces of Easily Covered Polytopes

Assume that $rB_{2}^{n} \subset P$ for some polytope $P \subset \mathbb{R}^n$, where $r \in (\frac{1}{2},1]$. Denote by $\mathcal{F}$ the set of facets of $P$, and by $N=N(P,B_2^n)$ the covering number of $P$ by the Euclidean unit ball $B_2^n$. We prove that if $\log N \le\frac{n}{8}$, then \[ |\mathcal{F}| \ge \left( \frac{1}{ 2\left(1 - r \sqrt{1-\frac{4\log N}{n}}\right) } \right)^{\frac{n-1}{2}}. \]

math.MG↗
arXiv2019

Functional Brunn-Minkowski Inequalities Induced by Polarity

We prove a new family of inequalities, which compare the integral of a geometric convolution of non-negative functions with the integrals of the original functions. For classical inf-convolution, this type of inequality is called the Prékopa-Leindler inequality, which, restricted to indicators of convex bodies, gives the classical Brunn-Minkowski inequality. The convolution we consider is a different one, which arises from the study of the polarity transform for functions. While inf-convolution arises as the pull back of usual addition of convex functions under the Legendre transform, our geometric inf-convolution arises as the pull back of the second order reversing transform on geometric convex functions (called either polarity transform or $A$-transform). These are, up to linear terms, the only order reversing isomorphisms on the class of geometric convex functions. We prove that the integral of this new geometric convolution of two functions is bounded from below by the harmonic average of the individual integrals. Our inequality implies the Brunn-Minkowski inequality, as well as some other, new, inequalities for volumes of bodies. Our inequalities are intimately connected with Busemann's convexity theorem, a new variant of which we prove for 1-convex hulls and log-concave densities.

math.FA↗
arXiv2019

A Santaló-type Inequality for the ${\cal J}$ Transform

This paper deals with an analog of the Mahler volume product related to the ${\cal J}$ transform acting in the class of geometric convex functions ${\rm{Cvx}}_0({\mathbb R}^n)$. We provide asymptotically sharp bounds for the quantity $s^{\cal J}(f) = \frac{\int e^{-{\cal J} f}}{\int e^{-f}}$ and characterize all the extremal functions.

math.FA↗
arXiv2019

On the Linear Structures Induced by the Four Order Isomorphisms Acting on ${\rm{Cvx}}_0({\mathbb R}^n)$

It is known that the volume functional $\,ϕ\mapsto\int e^{-ϕ}\,$ satisfies certain concavity or convexity inequalities with respect to three of the four linear structures induced by the order isomorphisms acting on ${\rm{Cvx}}_0({\mathbb R}^n)$. In this note we define the fourth linear structure on ${\rm{Cvx}}_0({\mathbb R}^n)$ as the pullback of the standard linear structure under the ${\cal J}$ transform. We show that, interpolating with respect to this linear structure, no concavity or convexity inequalities hold, and prove that a quasi-convexity inequality is violated only by up to a factor of $2$. We also establish all the order relations which the four different interpolations satisfy.

math.FA↗