A generalization of L-Brunn–Minkowski inequalities and L-Minkowski problems for measures

Stability of the Borell–Brascamp–Lieb Inequality for Multiple Power Concave Functions

“…and the continuity of u i , i = 1, • • • , m yields the continuity of u p,λ . For more details on supremal convolution of convex/concave functions, see [26][27][28][29].…”

“…Since the set u p,λ ≥ L λ = m ∑ i=1 {u i ≥ L i } is a single point set under the assumption that the involved functions are strictly p-concave, we then have that u p,λ ≥ L λ has zero measure. Then, using (26) and the fact µ i (s i ( t)) ≤ |Ω i |, and then using ( 28) and ( 15), we have…”

Lp-Curvature Measures andLp,q-Mixed Volumes

“…According to Equation (10), the curvature function of L p is the Radon-Nikodym derivative of L p -surface area measure with respect to the spherical Lebesgue measure. The integral of L p -curvature function (raised to an appropriate power) over the unit sphere is the L p -affine surface area, which is an important research point of affine geometry and valuation theory, see, e.g., [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. The L p -Minkowski problem (see [2]) is a necessary and sufficient condition to find a given measure such that it is only the L p -surface area measure of a convex body.…”