Lucas Marín, Eduardo

Person:
Lucas Marín, Eduardo

Name

Lucas Marín, Eduardo

publication.page.department

Universidad de Murcia. Departamento de Matemáticas

Full item page

Search Results

Now showing 1 - 7 of 7

Open Access
On discrete Brunn-Minkowski and isoperimetric type inequalities
(Elsevier, 2022-01) Iglesias López, David; Lucas Marín, Eduardo; Yepes Nicolás, Jesús; Matemáticas
We show that the lattice point enumerator Gn(·) satisfies G tK + sL + (−1,[t + s])n 1/n ≥ tG (K)1/n + sG (L)1/n for any K, L ⊂ Rn bounded sets with integer points and all t, s ≥ 0. We also prove that a certain family of compact sets, extending that of cubes [−m, m]n, with m ∈ N, minimizes the functional Gn(K + t[−1, 1]n), for any t ≥ 0, among those bounded sets K ⊂ Rn with given positive lattice point enumerator. Finally, we show that these new discrete inequalities imply the cor- responding classical Brunn-Minkowski and isoperimetric inequalities for non-empty compact sets.
Open Access
A discrete approach to Zhang’s projection inequality
(Cambridge University Press. Canadian Mathematical Society, 2025-09-24) Alonso Gutiérrez, David; Lucas Marín, Eduardo; Martín Goñí, Javier; Matemáticas
In this paper we will provide a new proof of the fact that for any convex body K ⊆ Rn 2n n n nn∞rn−1voln(K ∩ (ren + K))dr ≤0(voln(K))n+1,(voln−1(P ⊥ (K)))nn where (ei)n denotes the canonical orthonormal basis in Rn, P ⊥ (K) denotes n the orthogonal projection of K onto the linear hyperplane orthogonal to en, and volk denotes the k-dimensional Lebesgue measure. This inequality was proved by Gardner and Zhang and it implies Zhang’s inequality. We will use our new approach to this inequality in order to prove discrete analogues of this inequality and of an equivalent version of it, where we will consider the lattice point enumerator measure instead of the Lebesgue measure, and show that from such discrete analogues we can recover the aforementioned inequality and, therefore, Zhang’s inequality.
Open Access
On Rogers-Shephard type inequalities for thelattice point enumerator
(World Scientific Publishing, 2022-02-25) Alonso Gutiérrez, David; Lucas Marín, Eduardo; Yepes Nicolás, Jesús; Matemáticas
Abstract. In this paper we study various Rogers-Shephard type inequalities for the lattice point enumerator Gn(·) on Rn. In particular, for any non-empty convex bounded sets K,L ⊂ Rn. Additionally, a discrete counterpart to a classical result by Berwald for concave functions, from which other discrete Rogers-Shephard type inequalities may be derived, is shown. Furthermore, we prove that these new discrete analogues for Gn(·) imply the corresponding results involving the Lebesgue measure.
Open Access
On discrete Lp Brunn–Minkowski type inequalities
(Springer, 2022-08-11) Hernández Cifre, María de los Ángeles; Lucas Marín, Eduardo; Yepes Nicolás, Jesús; Matemáticas; Facultades de la UMU::Facultad de Matemáticas
Lp Brunn–Minkowski type inequalities for the lattice point enumerator Gn(·) are shown, p ≥ 1, both in a geometrical and in a functional setting. In particular, we prove that Gn((1 − λ) · K +p λ · L + (−1, 1)^n)^{p/n} ≥ (1 −λ)Gn(K)^{p/n} + λGn(L)^{p/n} for any K, L ⊂ R^n bounded sets with integer points and all λ ∈ (0, 1). We also show that these new discrete analogues (for Gn(·)) imply the corresponding results concerning the Lebesgue measure.
Open Access
On a characterization of lattice cubes via discrete isoperimetric inequalities
(Electronic Journal of Combinatorics, 2023-01-27) Lucas Marín, Eduardo; Iglesias López, David; Matemáticas
We obtain a characterization of lattice cubes as the only sets that reach equality in several discrete isoperimetric-type inequalities associated with the L∞ norm, including well-known results by Radcliffe and Veomett. We furthermore provide a new isoperimetric inequality for the lattice point enumerator that generalizes previous results, and for which the aforementioned characterization also holds.
Open Access
Interpolating between volume and lattice point enumerator with successive minima
(Springer, 2022-05-11) Freyer, Ansgar; Lucas Marín, Eduardo; Matemáticas; Facultad de Matemáticas
We study inequalities that simultaneously relate the number of lattice points, the volume and the successive minima of a convex body to one another. One main ingredient in order to establish these relations is Blaschke’s shaking procedure, by which the problem can be reduced from arbitrary convex bodies to anti-blocking bodies. As a consequence of our results, we obtain an upper bound on the lattice point enumerator in terms of the successive minima, which is equivalent to Minkowski’s upper bound on the volume in terms of the successive minima.
Open Access
On discrete log-Brunn-Minkowski type inequalities
(Society for Industrial and Applied Mathematics, 2022) Hernández Cifre, María de los Ángeles; Lucas Marín, Eduardo; Matemáticas
The conjectured log-Brunn-Minkowski inequality says that the volume of centrally symmetric convex bodies K, L ⊂ Rn satisfies vol (1−λ)·K+0 λ·L ≥ vol(K)1−λvol(L)λ, λ ∈ (0, 1), and is known to be true in the plane and for particular classes of n-dimensional symmetric convex bodies. In this paper we get some discrete log-Brunn-Minkowski type inequalities for the lattice point enumerator. Among others, we show that if K, L ⊂ Rn are unconditional convex bodies and λ ∈ (0, 1), then G (1 − λ) · K +C + λ · L+C 1 1 n ≥ G (K)1−λG (L)λ, where Cn = [−1/2, 1/2]n. Neither Cn nor (−1/2, 1/2)n can be removed. Furthermore, it implies the (volume) log-Brunn-Minkowski inequality for unconditional convex bodies. The corresponding results in the Lp setting for 0 < p < 1 are also obtained.