Bernardo González Merino

activity

20132026

most citedA generalization of the discrete version of Minkowski's fundamental theorem

9 citations · 26 across the 21 of their papers we have counts for

collaborators

31 papers

math.MG2026

Tight Stability Estimates Near the Simplex and Applications for the Banach-Mazur Distance and Rogers-Shephard-Type Inequalities

René Brandenberg, Bernardo González Merino, Florian Grundbacher

We establish a tight stability estimate for the Minkowski asymmetry near its maximal value, improving earlier results in both the range for the admissible error and the strength of…

math.MG2023

From inequalities relating symmetrizations of convex bodies to the diameter-width ratio for complete and pseudo-complete convex sets

René Brandenberg, Katherina von Dichter, Bernardo González Merino

For a Minkowski centered convex compact set $K$ we define $α (K)$ to be the smallest possible factor to cover $K \cap (- K)$ by a rescalation of $conv (K \cup (- K))$ and give…

math.MG2022

On diversities and finite dimensional Banach spaces

Bernardo González Merino

A diversity $δ$ in $M$ is a function defined over every finite set of points of $M$ mapped onto $[0, \infty)$ , with the properties that $δ (X) = 0$ if and only if $∣ X ∣ \leq 1$ and $δ(X\…

math.MG2021

Tightening and reversing the arithmetic-harmonic mean inequality for symmetrizations of convex sets

René Brandenberg, Katherina von Dichter, Bernardo González Merino

This paper deals with four symmetrizations of a convex set $C$ : the intersection, the harmonic and the arithmetic mean, and the convex hull of $C$ and $- C$ . A well-known result of…

math.MG2021★ 1 cited

On $(r, D, R)$ -Blaschke-Santaló diagrams with regular $k$ -gon gauges

René Brandenberg, Bernardo González Merino

We provide a study of Blaschke-Santaló diagrams for the inradius, diameter, and circumradius, measured with respect to different gauges. This contrasts previous works on those diag…

math.MG2020★ 2 cited

Large signed subset sums

Gergely Ambrus, Bernardo González Merino

We study the following question: for given $d \geq 2$ , $n \geq d$ and $k \leq n$ , what is the largest value $c (d, n, k)$ such that from any set of $n$ unit vectors in $R^{d}$ , w…

math.FA2020

Stone-Weierstrass theorem for homogeneous polynomials and its role in convex geometry

Bernardo González Merino, Rafael Villa

We give a uniform approximation of the characteristic function of the boundary of a centrally symmetric n-dimensional compact and convex set by homogeneous polynomials of even degr…

math.FA2020

Best approximation of functions by log-polynomials

David Alonso-Gutiérrez, Bernardo González Merino, Rafael Villa

Lasserre [La] proved that for every compact set $K \subset R^{n}$ and every even number $d$ there exists a unique homogeneous polynomial $g_{0}$ of degree $d$ with $K\subset G_1(…

math.MG2020★ 5 cited

Central diagonal sections of the $n$ -cube

Ferenc Bartha, Ferenc Fodor, Bernardo González Merino

We prove that the volume of central hyperplane sections of a unit cube in $R^{n}$ orthogonal to a diameter of the cube is a strictly monotonically increasing function of the…

math.MG2020

Relating Symmetrizations of Convex Bodies: Once More the Golden Ratio

René Brandenberg, Katherina von Dichter, Bernardo González Merino

We show that for any Minkowski centered planar convex compact set $C$ the Harmonic mean of $C$ and $- C$ can be optimally contained in the arithmetic mean of the same sets if and on…

math.FA2019

An extension of Berwald's inequality and its relation to Zhang's inequality

David Alonso-Gutiérrez, Julio Bernués, Bernardo González Merino

In this note prove the following Berwald-type inequality, showing that for any integrable log-concave function $f : R^{n} \to [0, \infty)$ and any concave function $h:L\ri…

math.FA2019★ 1 cited

On a generalization of the Hermite-Hadamard inequality and applications in convex geometry

Bernardo González Merino

In this paper we show the following result: if C is an n-dimensional 0-symmetric convex compact set, $f : C \to [0, 1)$ is concave, and $g : [0, 1) \to [0, 1)$ is not identica…

math.FA2019

On inscribed equilateral simplices in normed spaces

Bernardo González Merino

In this paper we prove in certain n-dimensional normed spaces X the existence of full-dimensional equilateral simplices of large size inscribed to the unit ball B. This extends the…

math.FA2019

Some remarks on Petty projection of log-concave functions

Leticia Alves da Silva, Bernardo González Merino, Rafael Villa

In this note we study the Petty projection of a log-concave function, which has been recently introduced in [9]. Moreover, we present some new inequalities involving this new notio…

math.MG2019

On the isodiametric and isominwidth inequalities for planar bisections

Antonio Cañete, Bernardo González Merino

For a given planar convex compact set $K$ , consider a bisection ${A, B}$ of $K$ (i.e., $A \cup B = K$ and whose common boundary $A \cap B$ is an injective continuous curve connecting…

math.FA2018

Zhang's inequality for log-concave functions

David Alonso-Gutiérrez, Julio Bernués, Bernardo González Merino

Zhang's reverse affine isoperimetric inequality states that among all convex bodies $K \subseteq R^{n}$ , the affine invariant quantity $∣ K ∣^{n - 1} ∣ Π^{*} (K) ∣$ (where $Π^{*} (K)$ deno…

math.MG2018

Uniqueness of circumcenters in generalized Minkowski spaces

Bernardo González Merino, Thomas Jahn, Christian Richter

In an $n$ -dimensional normed space every bounded set has a unique circumball if and only if every set of cardinality two has a unique circumball and if and only if the unit ball of…

math.MG2018

On the reverse isodiametric problem and Dvoretzky-Rogers-type volume bounds

Bernardo González Merino, Matthias Schymura

The isodiametric inequality states that the Euclidean ball maximizes the volume among all convex bodies of a given diameter. We are motivated by a conjecture of Makai Jr.~on the re…

math.MG2017★ 4 cited

Hunting for reduced polytopes

Bernardo González Merino, Thomas Jahn, Alexandr Polyanskii +1

We show that there exist reduced polytopes in three-dimensional Euclidean space. This partially answers the question posed by Lassak on the existence of reduced polytopes in $d$ -di…

math.MG2016

Minkowski concentricity and complete simplices

René Brandenberg, Bernardo González Merino

This paper considers the radii functionals (circumradius, inradius, and diameter) as well as the Minkowski asymmetry for general (possibly non-symmetric) gauge bodies. A generaliza…

math.MG2016

On densities of lattice arrangements intersecting every i-dimensional affine subspace

Bernardo González Merino, Matthias Henze

In 1978, Makai Jr. established a remarkable connection between the volume-product of a convex body, its maximal lattice packing density and the minimal density of a lattice arrange…

math.CO2016

Tight bounds on discrete quantitative Helly numbers

Gennadiy Averkov, Bernardo González Merino, Matthias Henze +2

Given a subset S of R^n, let c(S,k) be the smallest number t such that whenever finitely many convex sets have exactly k common points in S, there exist at most t of these sets tha…

math.MG2016

Is a complete, reduced set necessarily of constant width?

René Brandenberg, Bernardo González Merino, Thomas Jahn +1

Is it true that a convex body $K$ being complete and reduced with respect to some gauge body $C$ is necessarily of constant width, that is, satisfies $K - K = ρ (C - C)$ for some $ρ > 0$ ? W…

math.MG2016

Improving bounds for the Perel'man-Pukhov quotient for inner and outer radii

Bernardo González Merino

In this work we study upper bounds for the ratio of successive inner and outer radii of a convex body K. This problem was studied by Perel'man and Pukhov and it is a natural genera…

Tight Stability Estimates Near the Simplex and Applications for the Banach-Mazur Distance and Rogers-Shephard-Type Inequalities

From inequalities relating symmetrizations of convex bodies to the diameter-width ratio for complete and pseudo-complete convex sets

On diversities and finite dimensional Banach spaces

Tightening and reversing the arithmetic-harmonic mean inequality for symmetrizations of convex sets

On (r,D,R)-Blaschke-Santaló diagrams with regular k-gon gauges

Large signed subset sums

Stone-Weierstrass theorem for homogeneous polynomials and its role in convex geometry

Best approximation of functions by log-polynomials

Central diagonal sections of the n-cube

Relating Symmetrizations of Convex Bodies: Once More the Golden Ratio

An extension of Berwald's inequality and its relation to Zhang's inequality

On a generalization of the Hermite-Hadamard inequality and applications in convex geometry

On inscribed equilateral simplices in normed spaces

Some remarks on Petty projection of log-concave functions

On the isodiametric and isominwidth inequalities for planar bisections

Zhang's inequality for log-concave functions

Uniqueness of circumcenters in generalized Minkowski spaces

On the reverse isodiametric problem and Dvoretzky-Rogers-type volume bounds

Hunting for reduced polytopes

Minkowski concentricity and complete simplices

On densities of lattice arrangements intersecting every i-dimensional affine subspace

Tight bounds on discrete quantitative Helly numbers

Is a complete, reduced set necessarily of constant width?

Improving bounds for the Perel'man-Pukhov quotient for inner and outer radii

On $(r, D, R)$ -Blaschke-Santaló diagrams with regular $k$ -gon gauges

Central diagonal sections of the $n$ -cube