Inextensible domains

We develop a theory of planar, origin-symmetric, convex domains that are inextensible with respect to lattice covering, that is, domains such that augmenting them in any way allows fewer domains to cover the same area. We show that origin-symmetric inextensible domains are exactly the origin-symmetric convex domains with a circle of outer billiard triangles. We address a conjecture by Genin and Tabachnikov about convex domains, not necessarily symmetric, with a circle of outer billiard triangles, and show that it follows immediately from a result of Sas.Comment: Final submitted manuscript. Geometriae Dedicata, 201

Wolff-Denjoy theorems in non-smooth convex domains

We give a short proof of Wolff-Denjoy theorem for (not necessarily smooth) strictly convex domains. With similar techniques we are also able to prove a Wolff-Denjoy theorem for weakly convex domains, again without any smoothness assumption on the boundary.Comment: 13 page

Convex domains and K-spectral sets

Let $\Omega$ be an open convex domain of the complex plane. We study constants K such that $\Omega$ is K-spectral or complete K-spectral for each continuous linear Hilbert space operator with numerical range included in $\Omega$. Several approaches are discussed.Comment: the introduction was changed and some remarks have been added. 26 pages ; to appear in Math.

Some higher order isoperimetric inequalities via the method of optimal transport

In this paper, we establish some sharp inequalities between the volume and the integral of the $k$-th mean curvature for $k+1$-convex domains in the Euclidean space. The results generalize the classical Alexandrov-Fenchel inequalities for convex domains. Our proof utilizes the method of optimal transportation.Comment: 21 page

"Convex" characterization of linearly convex domains

We prove that a $C^{1,1}$-smooth bounded domain $D$ in \C^n is linearly convex if and only if the convex hull of any two discs in $D$ with common center lies in $D.$Comment: to appear in Math. Scand.; v3: Appendix is adde