In this paper we consider a family of convex sets in Rn, F = {S(x, t)}, x ∈ Rn, t > 0, satisfying certain axioms of affine invariance, and a Borel measure μ satisfying a doubling condition with… (More)

In this paper we establish several geometric properties of the cross sections of generalized solutions φ to the Monge-Ampère equation detD2φ = μ, when the measure μ satisfies a doubling property. A… (More)

Let φ be a convex function on a strictly convex domain Ω ⊂ Rn, n ≥ 1. The corresponding linearized Monge–Ampère equation is trace(ΦD2u) = f , where Φ := det D2φ (D2φ)−1 is the matrix of cofactors of… (More)

Let Ω ⊂ Rn be a bounded convex domain and φ ∈ C(Ω) be a convex function such thatφ is sufficiently smooth on∂Ω and the Monge–Ampère measure det D2φ is bounded away from zero and infinity in Ω. The… (More)

We prove local C1,α estimates of solutions for the parallel refractor and reflector problems under local assumptions on the target set Σ, and no assumptions are made on the smoothness of the… (More)

The problem considered in this paper is the following. Suppose we have a domain Ω ⊂ Rn−1 and a domain Σ contained in an n − 1 dimensional surface in Rn; Σ is referred to as the target domain or… (More)

where ν is the outer normal to A at the point where the light ray hits A. Suppose that we have a light source located at O, and Ω,Ω∗ are two domains in the sphere Sn−1, f(x) is a positive function… (More)

We establish a covering lemma of Besicovitch type for metric balls in the setting of Hölder quasimetric spaces of homogenous type and use it to prove a covering theorem for measurable sets. For… (More)