Browsing by Subject "Isoperimetric inequality"

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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.