## Differences of convex compacta and metric spaces of convex compacta with applications: A survey.(English)Zbl 1002.49022

Demyanov, V. (ed.) et al., Quasidifferentiability and related topics. Dedicated to Prof. Franco Giannessi on his 65th birthday and to Prof. Diethard Pallaschke on his 60th birthday. Dordrecht: Kluwer Academic Publishers. Nonconvex Optim. Appl. 43, 263-296 (2000).
The authors give a nice survey about the different definitions of algebraic set differences. It is pointed out that the most definitions base essentially on the Minkowski duality, i.e., on the (bijective) correspondence $$\varphi$$ between the family $$Y_n$$ of all convex compact sets of $$\mathbb{R}^n$$ and the family $$P_n$$ of all sublinear functions $$p: \mathbb{R}^n\to \mathbb{R}$$.
Let $$A,B\in Y_n$$ and $$p_A,p_B\in P_n$$ be the associated support functions. Since the difference $$p_A- p_B$$ is not convex in general, a set difference between $$A$$ and $$B$$ can be expressed by $A\ominus B= \varphi^{-1}(C(p_A- p_B)),$ where $$C(q)$$ is a suitable convexification of the positively homogeneous function $$q: \mathbb{R}^n\to \mathbb{R}$$. So the special set differences depend on the choice of the special convexification of $$p_A- p_B$$.
In the first part of the paper, some well-known set differences are discussed and compared, especially:
$$*$$ the difference using the metric projection $$C(q)= \text{Pr}_\infty(q)$$ which is the solution of the optimization problem $\max\{q(x)- p(x)\mid\|x\|\leq 1\}\to\min,\quad p\in P_n.$ $$*$$ the $$*$$-difference where $$C(q)$$ is the greatest convex minorant of $$q$$. By this we get the well-known representation $A\ominus B= \{x\mid B+ x\subset A\}.$ $$*$$ the Demyanov difference where $$C(q)$$ is the Clarke upper derivative of $$q$$. Here we have the representation by the Clarke subdifferential according to $A\ominus B= \partial_{Cl}(p_A- p_B)(0).$ $$*$$ the exposed difference defined by $A\ominus B= \{\nabla p_A(u)- \nabla p_B(u)\mid u\in T_A- T_B\},$ where $$T_A$$ and $$T_B$$ are the sets of points where $$p_A$$ and $$p_B$$ are differentiable, i.e., where the faces $$A(u)$$ and $$B(u)$$ are singletons (hence exposed points).
$$*$$ the quasidifferential according to $A\ominus B= \text{cl co}\bigcup \{A(u)- B(u)\mid u\in\mathbb{R}^n, u\neq 0\}.$ In the second part of the paper, the authors present some applications of set-differences in nonsmooth analysis, especially regarding the representation of generalized subdifferentials of DC and quasidifferentiable functions, the approximation of linear set-valued mappings and the construction of suitable metrics (the Demyanov metric and the Bartels-Pallaschke metric) in the space $$Y_n$$. Convergence, continuity and differentiability properties of polyhedral-valued mappings with respect to these metrics in comparison with the Hausdorff metric are pointed out.
For the entire collection see [Zbl 0949.00047].

### MSC:

 49J52 Nonsmooth analysis 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces) 58C06 Set-valued and function-space-valued mappings on manifolds 49J53 Set-valued and variational analysis