On the framework of $L_{p}$ summations for functions

Abstract

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.

On Minkowski and Blaschke symmetrizations of functions and related applications

“…3.3], all quermassintegrals of f are finite provided f is bounded and ∞ 0 r n dµ f (r) < ∞. Thus for any result in this paper dealing with quermassintegrals, we will assume that the functions involved satisfy these conditions (see also [60,Sec. 5]).…”

“…There are several standard ways of inducing an algebraic structure on C α (R n ). The first definition we present can be found in [11,60].…”

“…These operations are also commutative, associative and satisfy homogeneity properties; we refer the reader to [11,52,60]…”

“…The case of α = −∞ in Definition 2.4 is justified since both of these notions of mean width coincide in the limit as α → −∞. We refer the reader to [11,58,59,60] for the details.…”

“…This work represents another contribution to the (now vast) "probabilization of geometry" program, a term coined by V. Milman, which has been studied extensively during the past couple of decades in convex geometry and functional analysis. For some examples of the results in this program most closely related to this paper, we refer the reader to [1,11,23,24,25,39,52,53,58,59,60].…”