### A C1 function which is nowhere strongly paraconvex and nowhere semiconcave

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We give a complete characterization of closed sets $F\subset {\mathbb{R}}^{2}$ whose distance function ${d}_{F}:=\mathrm{dist}(\xb7,F)$ is DC (i.e., is the difference of two convex functions on ${\mathbb{R}}^{2}$). Using this characterization, a number of properties of such sets is proved.

In [2] a delta convex function on ${\mathbb{R}}^{2}$ is constructed which is strictly differentiable at $0$ but it is not representable as a difference of two convex function of this property. We improve this result by constructing a delta convex function of class ${C}^{1}\left({\mathbb{R}}^{2}\right)$ which cannot be represented as a difference of two convex functions differentiable at 0. Further we give an example of a delta convex function differentiable everywhere which is not strictly differentiable at 0.

P. Albano and P. Cannarsa proved in 1999 that, under some applicable conditions, singularities of semiconcave functions in ${\mathbb{R}}^{n}$ propagate along Lipschitz arcs. Further regularity properties of these arcs were proved by P. Cannarsa and Y. Yu in 2009. We prove that, for $n=2$, these arcs are very regular: they can be found in the form (in a suitable Cartesian coordinate system) $\psi \left(x\right)=(x,{y}_{1}\left(x\right)-{y}_{2}\left(x\right))$, $x\in [0,\alpha ]$, where ${y}_{1}$, ${y}_{2}$ are convex and Lipschitz on $[0,\alpha ]$. In other words: singularities propagate along arcs with finite turn.

In the present note we consider the definitions and properties of locally pseudo- and quasiconvex functions and give a sufficient condition for a locally quasiconvex function at a point x ∈ Rn, to be also locally pseudoconvex at the same point.

A strongly pseudoconvex function is generalized to non-smooth settings. A complete characterization of the strongly pseudoconvex radially lower semicontinuous functions is obtained.

We prove an abstract selection theorem for set-valued mappings with compact convex values in a normed space. Some special cases of this result as well as its applications to separation theory and Hyers-Ulam stability of affine functions are also given.

The computation of glacier movements leads to a system of nonlinear partial differential equations. The existence and uniqueness of a weak solution is established by using the calculus of variations. A discretization by the finite element method is done. The solution of the discrete problem is proved to be convergent to the exact solution. A first simple numerical algorithm is proposed and its convergence numerically studied.

The class of all functions f:(0,∞) → (0,∞) which are continuous at least at one point and affine with respect to the logarithmic mean is determined. Some related results concerning the functions convex with respect to the logarithmic mean are presented.