We study the exterior and interior Bernoulli problems for half Laplacian problem spectral Laplacian. concentrate on existence geometric properties of solutions. Our main results are following. For Laplacian, we show that under starshapedness assumptions data free domain is starshaped. convexity convex prove a Brunn–Minkowski inequality constant. use variational methods in combination with regul...

This note is devoted to the study of the dependence on p of the constant in the reverse Brunn-Minkowski inequality for p-convex balls (i.e. p-convex symmetric bodies). We will show that this constant is estimated as c ≤ C(p) ≤ C , for absolute constants c > 1 and C > 1. Let K ⊂ IR n and 0 < p ≤ 1. K is called a p-convex set if for any λ, μ ∈ (0, 1) such that λ + μ = 1 and for any points x, y ∈ ...

We initiate a systematic investigation into the nature of the function αK (L, ρ) that gives the volume of the intersection of one convex body K in Rn and a dilatate ρL of another convex body L in Rn, as well as the function ηK (L, ρ) that gives the (n − 1)-dimensional Hausdorff measure of the intersection of K and the boundary ∂(ρL) of ρL. The focus is on the concavity properties of αK (L, ρ). ...

Abstract An approach to interpolation of compact subsets $${{\mathbb {C}}}^n$$ C n , including Brunn–Minkowski type inequalities for the capacities interpolating sets, was developed in [8] by means plurisubharmonic geodesics between relative extre...

In this paper, we first introduce a new concept of dual quermassintegral sum function of two star bodies and establish Minkowski's type inequality for dual quermassintegral sum of mixed intersection bodies, which is a general form of the Minkowski inequality for mixed intersection bodies. Then, we give the Aleksandrov– Fenchel inequality and the Brunn–Minkowski inequality for mixed intersection...