Smooth exhaustion functions in convex domains

Convex Defining Functions for Convex Domains

We give three proofs of the fact that a smoothly bounded, convex domain in R n has defining functions whose Hessians are non-negative definite in a neighborhood of the boundary of the domain.

Quasiconformal Harmonic Functions between Convex Domains

We generalize Martio’s paper [14]. Indeed the problem studied in this paper is under which conditions on a homeomorphism f between the unit circle S1 := {z : |z| = 1} and a fix convex Jordan curve γ the harmonic extension of f is a quasiconformal mapping. In addition, we give some results for some classes of harmonic diffeomorphisms. Further, we give some results concerning harmonic quasiconfor...

Smooth Convex Bodies with Proportional Projection Functions

For a convex body K ⊂ Rn and i ∈ {1, . . . , n− 1}, the function assigning to any i-dimensional subspace L of Rn, the i-dimensional volume of the orthogonal projection of K to L, is called the i-th projection function of K. Let K, K0 ⊂ Rn be smooth convex bodies of class C2 +, and let K0 be centrally symmetric. Excluding two exceptional cases, we prove that K and K0 are homothetic if they have ...

On Non-smooth Convex Distance Functions

Optimization of Smooth and Strongly Convex Functions

A. Proof of Lemma 1 We need the following lemma that characterizes the property of the extra-gradient descent. Lemma 8 (Lemma 3.1 in (Nemirovski, 2005)). Let Z be a convex compact set in Euclidean space E with inner product 〈·, ·〉, let ‖ · ‖ be a norm on E and ‖ · ‖∗ be its dual norm, and let ω(z) : Z 7→ R be a α-strongly convex function with respect to ‖ · ‖. The Bregman distance associated wi...