Unified treatment of algebraic and geometric difference by a new difference scheme and its continuity properties (Q1810846)

The author introduces a new type of difference, depending on a parameter \(\alpha\geq 0\) between two sets \(A,B\) in \(\mathbb{R}^n\) in the following way \(:A\overset{\alpha}{-} C=\{x\in\mathbb{R}^n:\mathcal{L}((x+C)\cap A)\geq \alpha\}\), where \(\mathcal{L}\) denotes the Lebesgue measure. It is clear that, if \(\alpha>\min(\mathcal{L}(C),\mathcal{L}(A))\) then \(A\overset{\alpha}{-} C\not=\emptyset\), and for this reason, in the paper one supposes \(\alpha\leq \min(\mathcal{L}(C),\mathcal{L}(A))\). If \(A- C\) is the classical difference, and \(A\overset{*}{-} C\) the Minkowski difference (i.e. \(A\overset{*}{-} C=\{x\in\mathbb{R}^n:x+C\subset A\}\), then these operations and the difference \(\overset{\alpha}{-}\) are linked by the following properties: (i) \(A-C=A\overset{0}{-} C\); (ii) if \(0\leq\alpha\leq\mathcal{L}(C)\), then \(A\overset{*}{-} C\subset A\overset{\mathcal{L}(C)}{-} C\subset A\overset{\alpha}{-} C\subset A-C\); (iii) if \(A\) and \(C\) are closed convex sets in \(\mathbb{R}^n\), at least one bounded, and \(\operatorname{int}C\not=\emptyset\) then \(A\overset{*}{-} C=A\overset{\mathcal{L}(C)}{-} C\). Next the author proves that the difference \(\,\overset{\alpha}{-}\,\) is continuous with respect to both the parameter \(\alpha\) and the operands. The last section is devoted to Lipschitz properties of the Minkowski difference in a normal vector space.