Browsing by Subject "Cardinality"

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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.