Characterization of bounded sets in terms of asymptotic cones and homotheties (Q2192142)

The author presents characterizations of bounded sets in a Banach space \(X\), based on the asymptotic cone defined for a subset \(A\subset X\) as \(A_{\infty}=\{x\in X:\exists t_{n}>0\), \(t_{n}\rightarrow 0\), \(\exists x_{n}\in A\), \(\lim_{n\rightarrow \infty }t_{n}x_{n}=x\}\). The author introduces the Hausdorff deviation \(\beta (A,B)\) and the Hausdorff distance \(\alpha (A,B)\) between two nonempty subsets \(A\) and \(B\) through \(\beta (A,B)=\sup_{x\in A}\inf_{y\in B}\left\Vert x-y\right\Vert \) and \(\alpha (A,B)=\max (\beta(A,B),\beta (B,A))\). He defines the notion of asymptotically boundedness for a nonempty set \(A\) if \(\beta (A,A_{\infty })<+\infty \) and of regularly asymptotically boundedness if \(\alpha (A,A_{\infty })<+\infty \). He defines seven properties P\(_{0}\): \(\beta (A,A_{\infty })<+\infty \), P\(_{1}\): \(\exists \lambda \in (0,1)\) \(\beta (A,\lambda A)<+\infty \), P\(_{2}\): \(\lim\inf_{\lambda \rightarrow 0^{+}}\beta (A,\lambda A)<+\infty \), P\(_{3}\): \(\forall \lambda \in (0,1)\) \(\beta (A,\lambda A)<+\infty \), P\(_{4}\): \(\sup_{0<\lambda <1}\beta (A,\lambda A)<+\infty \), P\(_{5}\): \(\alpha (A,A_{\infty })<+\infty \), Q: \(\beta (A_{\infty },A)<+\infty \). The first main result proves that for every nonempty subset \(A\subset X\), the following three conditions are equivalent: \(A\) is bounded, the seven conditions \(A_{\infty }=\{0\}\), P\(_{0}\),\(\ldots \), P\(_{5}\) hold, the condition \(A_{\infty }=\{0\}\) and any of the conditions P\(_{0}\),\(\ldots \), P\(_{5}\) hold true. The second main result proves that for every subset \(A\) in the class \(\mathbf{K}_{0}=\{A\subset X:A\neq \varnothing \), \(A_{\infty }=\{0\}\}\), the following seven conditions are equivalent: \(A\) is bounded, P\(_{0}\),\(\dots \), P\(_{5}\). The third main result proves that for every subset \(A\) in the class \(\mathbf{K}_{1}=\{A\subset X:A\neq \varnothing \), \(\beta (A_{\infty },A)<+\infty \}\), the conditions P\(_{0}\),\(\dots \), P\(_{5}\) are equivalent. The author proves that these main results can be deduced from the following proposition: for any nonempty subset \(A\subset X\), the implications \(A\) is bounded \(\Rightarrow \) P\(_{5}\) \(\Rightarrow \) P\(_{4}\) \( \Rightarrow \) P\(_{3}\) \(\Rightarrow \) P\(_{2}\) \(\Leftrightarrow \) P\(_{1}\) \(\Rightarrow \) P\(_{0}\) hold true. For the proof of this proposition, the author draws computations and establishes bounds on \(\beta (A,\lambda A)\) and \(\beta (A,\lambda A+(1-\lambda )A_{\infty })\) for \(\lambda \in (0,1)\), especially in the case where \(A+A_{\infty }\subset \overline{A}\). The paper ends with the presentation of examples of subsets \(A\) which satisfy \(A+A_{\infty }\subset \overline{A}\), or which belong to the class \(\mathbf{K}_{1}\), and with examples in \(\mathbb{R}^{2}\) or \(\mathbb{R}\) which prove that some reverse implications in the last proposition are not true.