Abstract
We show that the lattice point enumerator Gn(⋅) satisfiesGn(tK+sL+(−1,⌈t+s⌉)n)1/n≥tGn(K)1/n+sGn(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 corresponding classical Brunn-Minkowski and isoperimetric inequalities for non-empty compact sets.
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