Browsing by Subject "Integer lattice"

Now showing 1 - 3 of 3
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    Brunn-Minkowski type inequalities for the lattice point enumerator
    (Elsevier, 2020-08-26) Iglesias, David; Zvavitch, Artem ; Yepes Nicolás, Jesús; Matemáticas; Facultades de la UMU::Facultad de Matemáticas
    Geometric and functional Brunn-Minkowski type inequalities for the lattice point enumerator Gn(⋅) are provided. In particular, we show that Gn((1−λ)K+λL+(−1,1)^n)^{1/n}≥(1−λ)Gn(K)^{1/n}+λGn(L)^{1/n} for any non-empty bounded sets K,L⊂R^n and all λ∈(0,1). We also show that these new discrete versions imply the classical results, and discuss some links with other related inequalities.
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    On discrete Borell–Brascamp–Lieb inequalities
    (EMS Press, 2019-09-27) Iglesias, David; Yepes Nicolás, Jesús; Matemáticas; Facultad de Matemáticas
    If f,g,h:R n ⟶R ≥0 are non-negative measurable functions such that h(x+y) is greater than or equal to the p-sum of f(x) and g(y), where −1/n≤p≤∞, p=/0, then the Borell–Brascamp–Lieb inequality asserts that the integral of h is not smaller than the q-sum of the integrals of f and g, for q=p/(np+1). In this paper we obtain a discrete analog for the sum over finite subsets of the integer lattice Z n : under the same assumption as before, for A,B⊂Z n }, then ∑ A+B h≥[(∑ rf(A) f) q +(∑ B g) q ] 1/q , where r f (A) is obtained by removing points from A in a particular way, and depending on f. We also prove that the classical Borell–Brascamp–Lieb inequality for Riemann integrable functions can be obtained as a consequence of this new discrete version.
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    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.