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Bernardo González Merino

Publications and source records attributed to Bernardo González Merino.

42 records · Page 3Linked to original sources
arXiv2016

A generalization of the discrete version of Minkowski's fundamental theorem

One of the most fruitful results from Minkowski's geometric viewpoint on number theory is his so called 1st Fundamental Theorem. It provides an optimal upper bound for the volume of an o-symmetric convex body whose only interior lattice point is the origin. Minkowski also obtained a discrete analog by proving optimal upper bounds on the number of lattice points in the boundary of such convex bodies. Whereas the volume inequality has been generalized to any number of interior lattice points already by van der Corput in the 1930s, a corresponding result for the discrete case remained to be proven. Our main contribution is a corresponding optimal relation between the number of boundary and interior lattice points of an o-symmetric convex body. The proof relies on a congruence argument and a difference set estimate from additive combinatorics.

math.CO↗
arXiv2015

John's ellipsoid and the integral ratio of a log-concave function

We extend the notion of John's ellipsoid to the setting of integrable log-concave functions. This will allow us to define the integral ratio of a log-concave function, which will extend the notion of volume ratio, and we will find the log-concave function maximizing the integral ratio. A reverse functional affine isoperimetric inequality will be given, written in terms of this integral ratio. This can be viewed as a stability version of the functional affine isoperimetric inequality.

math.FA↗
arXiv2015

The asymmetry of complete and constant width bodies in general normed spaces and the Jung constant

In this paper we state a one-to-one connection between the maximal ratio of the circumradius and the diameter of a body (the Jung constant) in an arbitrary Minkowski space and the maximal Minkowski asymmetry of the complete bodies within that space. This allows to generalize and unify recent results on complete bodies and to derive a necessary condition on the unit ball of the space, assuming a given body to be complete. Finally, we state several corollaries, i.e. concerning the Helly dimension or the Banach-Mazur distance.

math.MG↗
arXiv2015

A pseudometric invariant under similarities in the hyperspace of non-degenerated compact convex sets of $\mathbb R^n$

In this work we define a new pseudometric in $\mathcal K^n_*$, the hyperspace of all non-degenerated compact convex sets of $\mathbb R^n$, which is invariant under similarities. We will prove that the quotient space generated by this pseudometric (which is the orbit space generated by the natural action of the group of similarities on $\mathcal K^n_*$) is homeomorphic to the Banach-Mazur compactum $BM(n)$, while $\mathcal K^n_*$ is homeomorphic to the topological product $Q\times\mathbb R^{n+1}$, where $Q$ stands for the Hilbert cube. Finally we will show some consequences in convex geometry, namely, we measure how much two convex bodies differ (by means of our new pseudometric) in terms of some classical functionals.

math.MG↗
arXiv2014

A complete 3-dimensional Blaschke-Santaló diagram

We present a complete 3-dimensional Blaschke-Santaló diagram for planar convex bodies with respect to the four classical magnitudes inner and outer radius, diameter and (minimal) width in euclidean spaces.

math.MG↗