Birandom variable is a generalization of random variable, which is defined as a measurable function from a probability space to the set of random variables. In order to further explore the mathematical properties of birandom variable, this paper first investigates several inequalities for birandom variables based on the chance measure and expected value operator, which are analogous to Hölder’s...

The classical Minkowski inequality in the Euclidean space provides a lower bound on total mean curvature of hypersurface terms surface area, which is optimal round spheres. In this paper we employ locally constrained inverse flow to prove properly defined analogue Lorentzian de Sitter space.

The Brunn-Miknowski inequality gives a lower bound on the Lebesgue measure of a sumset in terms of the measures of the individual sets. This classical inequality in convex geometry was inspired by issues around the isoperimetric problem and was considered for a long time to belong to geometry, where its significance is widely recognized. However, it is by now clear that the Brunn-Miknowski ineq...

We prove a quantitative stability result for the Brunn-Minkowski inequality: if |A| = |B| = 1, t ∈ [τ, 1−τ ] with τ > 0, and |tA+(1−t)B| ≤ 1+δ for some small δ, then, up to a translation, both A and B are quantitatively close (in terms of δ) to a convex set K.