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Bernardo González Merino

Publications and source records attributed to Bernardo González Merino.

At least 19 recordsLinked to original sources
arXiv2026

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

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 the estimate. More precisely, if an $n$-dimensional convex body $K$ has Minkowski asymmetry $s(K) \geq n-\varepsilon$ for $\varepsilon \in [0,1)$, then its Banach-Mazur distance to the $n$-simplex is at most \[ 1 + \varepsilon + \frac{\varepsilon^2}{2(1-\varepsilon)}. \] This dimension-independent estimate is sharp to the linear order in $\varepsilon$, including the constant. We apply this estimate to several problems. First, we prove a stability result for the maximal Banach-Mazur distance to the Euclidean ball, improving a previous estimate to the optimal linear order. As a key ingredient, we verify the conjecture that every convex body $K$ contains a translated copy of its volume-minimal circumscribed ellipsoid scaled down by a factor $\sqrt{n s(K)}$. Second, we prove a sharp common generalization of Schneider's higher-order Rogers-Shephard inequality and the $L_p$-Rogers-Shephard inequality, and establish a stability result of the optimal linear order. These results are based on the recent positive answer to the inequality part of the higher-order Godbersen conjecture and the accompanying proof of the $L_p$-Rogers-Shephard inequality. Finally, we improve upper bounds for the diameter of the Banach-Mazur compactum in fixed dimensions.

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arXiv2026

On the Perelman-Pukhov quotient of successive radii: better and asymptotically optimal bounds

Perel'man in 1987 and independently Pukhov in 1979 proved that the quotient between the $(n-i+1)$-th successive outer radius and the $i$-th successive inner radius of a convex body in $n$-dimensions is not larger than $i+1$. Apart from the solved cases by Jung 1901 $(i=1)$ and Steinhagen 1921 $(i=n)$, only Perel'man (1987, $n=3$, $i=2$) and González Merino (2017, $n\geq 4$, $i=2$ and $i=n-1$) provided small improvements that beat this bound. In this paper, we obtain sharper inequalities using relations between these inner and outer measures with the diameter and minimal width. We improve the current bounds in the following cases: $i=3$ when $4\leq n \leq 8$, $i=4$ when $n=5$, $6$, $i=5$ when $n=6$, $i=6$ when $n=7$, and for every $i\geq n-Θ(\log n)$. Notably, our bounds provide the right order in $n$ when $i=n-m$, with $m$ constant and $n$ arbitrarily large. Additionally, we improve the case $n=5$, $i=3$ even further by refining an idea of Perel'man and using the optimal lower bound of the inradius in terms of the circumradius and the diameter in 3-space (see [7]).

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arXiv2026

Minimization of the inradius of convex bodies for prescribed diameter and circumradius in Minkowski spaces

We study Blaschke-Santaló diagrams for the inradius, circumradius, and diameter in arbitrary-dimensional Minkowski spaces. Their boundary parts are regularly described by four linear inequalities and one more complex inequality, and we analyze which bodies fill the boundaries and for which (types of) norms that boundaries collapse to a single point. In this context, the diagram with respect to the 1-norm in 3-space plays an exceptional role. We complete the description of the according diagram by tightening Bohnenblust's inequality through involvement of the inradius.

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arXiv2026

Central diagonal sections of Gaussian cubes

The investigation of the volume, surface area, and other geometric properties of sections of convex bodies, and in particular cubes, has a long history and a rich literature. However, much less is known when the cube has a volume distribution that is different from the Lebesgue measure; for example, a Gaussian density. We study the probability densities in the standard cube $B^n_\infty=[-1,1]^n$ of $\mathbb R^n$ generated by $e^{-b\|x\|^2}$, $b> 0$. We prove that the limit of the induced Gaussian-type volume of hyperplane sections of $B^n_\infty$ through the origin and orthogonal to a main diagonal is \[ \sqrt{\frac bπ}\left (1-4\frac{e^{-b}\sqrt{b}}{2\sqrtπ\mathrm{erf}(\sqrt{b})}\right)^{-\frac12}, \] as $n\to\infty$. This extends the well-known result of Hensley (1979) for the Lebesgue measure and continues the investigations initiated by Barthe, Guédon, Mendelson, Naor (2005), Zvavitch (2008), and König, Koldobski (2013).

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arXiv2026

On isoperimetric local-Bollobás-Thomason inequalities

We prove the following isoperimetric-type inequality: for every convex body $K$ in $\mathbb R^n$ and some $σ\subset[n]:=\{1,\dots,n\}$ there exists a suitable Hanner polytope $B_K$ with the same volume as $K$ and such that the volume of each of its orthogonal projections onto every subspace whose basis is formed by the canonical vectors $\{e_i:i\inτ\cup([n]\setminusσ)\}$, for every $τ\subseteqσ$, bounds from below the volume of the corresponding projections of $K$.

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arXiv2025

On local Liakopoulos-Meyer type inequalities and their functional counterparts

We provide a functional Rogers-Shephard type inequality for log-concave functions on $\mathbb R^n$ and any $1$-reducible $s$-cover of $[n]$. As a consequence, we derive a sharp local Liakopoulos-Meyer type inequality for $n$-dimensional convex bodies and $1$-reducible $s$-covers of any $σ\subset[n]$, solving a question studied by Brazitikos, Giannopoulos, Liakopoulos in [14] as well as Alonso-Gutiérrez, Bernués, Brazitikos, Carbery in [3].

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arXiv2024

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

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 $\mathrm{conv} (K\cup (-K))$ and give a complete description of the possible values of $α(K)$ in the planar case in dependence of the Minkowski asymmetry of $K$. As a side product, we show that, if the asymmetry of $K$ is greater than the golden ratio, the boundary of $K$ intersects the boundary of its negative $-K$ always in exactly 6 points. As an application, we derive bounds for the diameter-width-ratio for pseudo-complete and complete sets, again in dependence of the Minkowski asymmetry of the convex bodies, tightening those depending solely on the dimension given in a recent result of Richter [10].

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arXiv2023

Narrowing the gaps of the missing Blaschke-Santaló diagrams

We solve several new sharp inequalities relating three quantities amongst the area, perimeter, inradius, circumradius, diameter, and minimal width of planar convex bodies. As a consequence, we narrow the missing gaps in each of the missing planar Blaschke-Santaló diagrams. Furthermore, we extend some of those sharp inequalities into higher dimensions, by replacing either the perimeter by the mean width or the area by the volume.

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arXiv2023

Some remarks on Petty projection of log-concave functions

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 notion, partly complementing and correcting some results from [9].

math.FA↗
arXiv2023

On diversities and finite dimensional Banach spaces

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 $δ(X\cup Y)\leqδ(X\cup Z)+δ(Z\cup Y)$, for every finite sets $X,Y,Z\subset M$ with $|Z|\geq 1$. Its importance relies in the fact that, amongst others, they generalize the notion of metric distance. Our main contribution is the characterization of Banach-embeddable diversities $δ$ defined over $M$, $|M|=3$, i.e. when there exist points $p_i\in\mathbb R^n$, $i=1,2,3$, and a symmetric, convex, and compact set $C\subset\mathbb R^n$ such that $δ(\{x_{i_1},\dots,x_{i_m}\})=R(\{p_{i_1},\dots,p_{i_m}\},C)$, where $R(X,C)$ denotes the circumradius of $X$ with respect to $C$.

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arXiv2022

Large signed subset sums

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 $\mathbb{R}^d$, we may select $k$ vectors with corresponding signs $\pm 1$ so that their signed sum has norm at least $c(d,n,k)$? The problem is dual to classical vector sum minimization and balancing questions, which have been studied for over a century. We give asymptotically sharp estimates for $c(d,n,k)$ in the general case. In several special cases, we provide stronger estimates: the quantity $c(d,n,n)$ corresponds to the $\ell_p$-polarization problem, while determining $c(d, n, 2)$ is equivalent to estimating the coherence of a vector system, which is a special case of $p$-frame energies. Two new proofs are presented for the classical Welch bound when $n = d+1$. For large values of $n$, volumetric estimates are applied for obtaining fine estimates on $c(d,n,2)$. Studying the planar case, sharp bounds on $c(2, n, k)$ are given. Finally, we determine the exact value of $c(d,d+1,d+1)$ under some extra assumptions.

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arXiv2022

Relating Symmetrizations of Convex Bodies: Once More the Golden Ratio

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 only if the Minkowski asymmetry of $C$ is at most the golden ratio $(1+\sqrt{5})/2 \approx 1.618$. Moreover, the most asymmetric such set that is (up to a linear transformation) a special pentagon, which we call the golden house.

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arXiv2021

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

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 Firey shows that those means build up a subset-chain in the given order. On the one hand, we determine the dilatation factors, depending on the asymmetry of $C$, to reverse the containments between any of those symmetrizations. On the other hand, we tighten the relations proven by Firey and show a stability result concerning those factors near the simplex.

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arXiv2021

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

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 diagrams, which are all considered for euclidean measure. By proving several new inequalities and properties between these three functionals, we compute the intersection and the union over all possible gauges of those diagrams, showing that they coincide with the corresponding diagrams of a parallelotope and (in the planar case) a triangle, respectively. Further new extremal properties are derived considering the diagrams with respect to a regular pentagon or hexagon gauge.

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arXiv2021

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

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 degree $d$ fulfilling $|g_d-1|\leq E/d^{1/2-β}$, for every $β>0$, large enough $d$, and some constant $E$ only depending on $n$ and $K$. In particular, this proves a conjecture posed by Kroo in 2004, also known as the Stone-Weierstrass theorem for homogeneous polynomials. Moreover, we introduce the d-volume ratio for a convex body $K$ in $\mathbb R^n$, by means of its d-Lasserre-Löwner polynomial. We also prove an upper bound of the d-volume ratio of the form $1+F/d^{3/2-β}$, for every $β>0$, large enough $d$, and $F$ some constant only depending on $n$.

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arXiv2021

Best approximation of functions by log-polynomials

Lasserre [La] proved that for every compact set $K\subset\mathbb R^n$ and every even number $d$ there exists a unique homogeneous polynomial $g_0$ of degree $d$ with $K\subset G_1(g_0)=\{x\in\mathbb R^n:g_0(x)\leq 1\}$ minimizing $|G_1(g)|$ among all such polynomials $g$ fulfilling the condition $K\subset G_1(g)$. This result extends the notion of the Löwner ellipsoid, not only from convex bodies to arbitrary compact sets (which was immediate if $d=2$ by taking convex hulls), but also from ellipsoids to level sets of homogeneous polynomial of an arbitrary even degree. In this paper we extend this result for the class of non-negative log-concave functions in two different ways. One of them is the straightforward extension of the known results, and the other one is a suitable extension with uniqueness of the solution in the corresponding problem and a characterization in terms of some 'contact points'.

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arXiv2021

Estimating the average of functions with convexity properties by means of a new center

In this article we show the following result: if $C$ is an $n$-dimensional convex and compact subset, $f:C\rightarrow[0,\infty)$ is concave, and $ϕ:[0,\infty)\rightarrow[0,\infty)$ is a convex function with $ϕ(0)=0$, we then characterize the class of sets and concave functions that attain the supremum \[ \sup_{C,f}\int_Cϕ(f(x))dx, \] where the supremum ranges over all sets $C$ with $n$-dimensional volume $|C|=c$ and the additional condition that $f(x_{C,f})=k$ for some point $x_{C,f}\in C$ that we introduce in the article, for two non-negative constants $c,k>0$. As a consequence, we extend some results of Milman and Pajor in [MP] and some in [Thm. 1.2, GoMe]. Besides, we also obtain some new estimates on the volume of particular sections of a convex set $K$ passing through a new point of $K$.

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