In this article we are interested in the following problem: to find a map $u:\Omega \to {ℝ}^2$ that satisfies$$\left\{\begin{array}{cc}Du\in E\hfill & \mathit{\text{a.e.}}\text{in}\Omega \hfill \\ u\left(x\right)=\varphi \left(x\right)\hfill & x\in \partial \Omega \hfill \end{array}\right.$$where $\Omega$ is an open set of ${ℝ}^2$ and $E$ is a compact isotropic set of ${ℝ}^{2\times 2}$. We will show an existence theorem under suitable hypotheses on $\varphi$.

In this article we are interested in the following problem: to find a map $u:\Omega \to {ℝ}^2$ that satisfies $$\left\{\begin{array}{cc}Du\in E\hfill & \mathit{\text{a.e.}}\text{in}\Omega \hfill \\ u\left(x\right)=\varphi \left(x\right)\hfill & x\in \partial \Omega \hfill \end{array}\right.$$ where Ω is an open set of ${ℝ}^2$ and E is a compact isotropic set of ${ℝ}^{2\times 2}$. We will show an existence theorem under suitable hypotheses on φ.

In the first part of the paper, we define an approximated Brunn-Minkowski inequality which generalizes the classical one for metric measure spaces. Our new definition, based only on properties of the distance, allows also us to deal with discrete metric measure spaces. Then we show the stability of our new inequality under convergence of metric measure spaces. This result gives as corollary the stability of the classical Brunn-Minkowski inequality for geodesic spaces. The proof of this stability...

Let $F$ be a subfield of the field $ℝ$ of real numbers. Equipped with the binary arithmetic mean operation, each convex subset $C$ of $F^n$ becomes a commutative binary mode, also called idempotent commutative medial (or entropic) groupoid. Let $C$ and $C^{\text{'}}$ be convex subsets of $F^n$. Assume that they are of the same dimension and at least one of them is bounded, or $F$ is the field of all rational numbers. We prove that the corresponding idempotent commutative medial groupoids are isomorphic iff the affine space $F^n$...

In this paper, we will give a new fixed point theorem for lower semicontinuous multimaps in a Hausdorff topological vector space.

Let K be a convex body in ${ℝ}^n$ and B be the Euclidean unit ball in ${ℝ}^n$. We show that $li{m}_{t\to 0}\left(\right|K|-|K_t\left|\right)/\left(\right|B|-|B_t\left|\right)=as\left(K\right)/as\left(B\right)$, where as(K) respectively as(B) is the affine surface area of K respectively B and ${K_t}_{t\ge 0}$, ${B_t}_{t\ge 0}$ are general families of convex bodies constructed from K,B satisfying certain conditions. As a corollary we get results obtained in [M-W], [Schm], [S-W] and [W].

By elementary geometric arguments, correlation inequalities for radially symmetric probability measures are proved in the plane. Precisely, it is shown that the correlation ratio for pairs of width-decreasing sets is minimized within the class of infinite strips. Since open convex sets which are symmetric with respect to the origin turn out to be width-decreasing sets, Pitt’s Gaussian correlation inequality (the two-dimensional case of the long-standing Gaussian correlation conjecture) is derived...