On Multiple Lₚ-Curvilinear-Brunn–Minkowski Inequalities

We construct the extension of the curvilinear summation for bounded Borel measurable sets to the Lₚ L p space for multiple power parameter α ̄=(α ₁, … , α ₙ₊₁) α ¯ = ( α 1 , … , α n + 1 ) when $$p>0$$ p > 0 . Based on this Lp,α ̄ L p , α ¯ -curvilinear summation for sets and the concept of compression of sets, the Lp,α ̄ L p , α ¯ -curvilinear-Brunn–Minkowski inequality for bounded Borel measurable sets and its normalized version are established. Furthermore, by utilizing the hypo-graphs for functions, we enact a brand new proof of Lp,α ̄ L p , α ¯ Borell–Brascamp–Lieb inequality, as well as its normalized version, for functions containing the special case of the Lₚ L p Borell–Brascamp–Lieb inequality through the Lp,α ̄ L p , α ¯ -curvilinear-Brunn–Minkowski inequality for sets. Moreover, we propose the multiple power Lp,α ̄ L p , α ¯ -supremal-convolution for two functions together with its properties. Last but not least, we introduce the definition of the surface area originated from the variation formula of measure in terms of the Lp,α ̄ L p , α ¯ -curvilinear summation for sets as well as Lp,α ̄ L p , α ¯ -supremal-convolution for functions together with their corresponding Minkowski type inequalities and isoperimetric inequalities for p≥ 1, p ≥ 1 , etc.