We develop the framework of L p operations for functions by introducing two primary new types L p,s summations for p > 0: the L p,s convolution sum and the L p,s Asplund sum for functions. The first type is defined as the linear summations of functions in terms of the L p coefficients (C p,λ,t , D p,λ,t ), the so-called the L p,s supremal-convolution when p ≥ 1 and the L p,s inf-sup-convolution when 0 < p < 1, respectively. The second type L p,s summation is created by the L p averages of bases for s-concave functions. We show that they are equivalent in the case s = 0 (log-concave functions) and p ≥ 1. For the former type L p,s summation, we establish the corresponding L p -Borell-Brascamp-Lieb inequalities for all s ∈ [−∞, ∞] and p ≥ 1. Furthermore, in summarizing the conditions for these new types of L p -Borell-Brascamp-Lieb inequalities, we define a series of the L p,s concavity definitions for functions and measures. On the other hand, for the latter type L p,s Asplund summation, we discover the integral formula for L p,s mixed quermassintegral for functions via tackling the variation formula of quermassintegral of functions for p ≥ 1.
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