Vitale, Richard A. \(L_p\) metrics for compact convex sets. (English) Zbl 0595.52005 J. Approximation Theory 45, 280-287 (1985). Let \(\mathcal K^d\) the space of non-empty convex compact subsets of \(\mathbb R^d\) \((2\leq d<+\infty)\). The Hausdorff metric \(\delta_\infty\) on \(\mathcal K^d\) is given by the formula \(\delta_\infty(K,L)=\sup_{e\in S^{d-1}}| \delta^*(e,K)-\delta^*(e,L)|\) where \(\delta^*(e,\cdot)\) is the support functional of a non-empty convex compact set in \(\mathbb R^d\) and \(S^{d-1}\) the unit sphere. The author establishes some results relating the \(\delta_{\infty}\) metric with the \(\delta_p\) metric on \(\mathcal K^d\) defined by \(\delta_p(K,L)=[\int_{S^{d-1}}| \delta^*(e,K)-\delta^*(e,L)|^p \mu(de)]^{1/p}\) where \(\mu\) is unit Lebesgue measure on \(S^{d-1}\) and \(1\leq p<+\infty\). Theorem 1: Let \(K, L\in\mathcal K^d\). Then \(\delta_p(K,L)\leq \delta_\infty(K,L)\). Equality is attained iff one set is a parallel body of the other. The main result is Theorem 2 (p. 282) giving tight bounds between the \(\delta_p\) \((1\leq p<+\infty)\) and \(\delta_\infty\) metrics. As consequence of Theorem 1 and Theorem 2, the author proves that \((\mathcal K^d,\delta_p)\) \((1\leq p\leq \infty)\) are complete in which closed bounded sets are compact, and all the metrics \(\delta_p\) \((1\leq p\leq \infty)\) induce the same topology on \(\mathcal K^d\). Reviewer: C. Castaing (Montpellier) Cited in 49 Documents MSC: 52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces) 52A22 Random convex sets and integral geometry (aspects of convex geometry) 41A99 Approximations and expansions 46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) Keywords:approximation of convex sets; \(L_p\) metrics; \(L_\infty\) metrics PDFBibTeX XMLCite \textit{R. A. Vitale}, J. Approx. Theory 45, 280--287 (1985; Zbl 0595.52005) Full Text: DOI References: [1] Artstein, Z., Limit Laws for Multifunctions Applied to an Optimization Problem, (Technical report (1984), Weizmann Institute of Science) · Zbl 0563.60032 [2] Baddeley, A., Stochastic geometry: An introduction and reading-list, Internat. Statist. Rev., 50, 179-193 (1982) · Zbl 0503.60018 [3] Davis, P. J., Lemoine Approximation and Steiner Approximation of Convex Sets, (Technical report (1982), Brown University) [4] Davis, P. J.; Vitale, R. A.; Ben-Sabar, E., On the deterministic and stochastic approximation of regions, J. Approx. Theory, 21, 60-88 (1977) · Zbl 0355.41032 [5] Firey, W. J., Polar means of convex bodies and a dual to the Brunn-Minkowski theorem, Canad. J. Math., 13, 444-453 (1961) · Zbl 0126.18005 [6] Gruber, P. M., Approximation of convex bodies, (Gruber, P. M.; Wills, J. M., Convexity and Its Applications (1983), Birkhäuser: Birkhäuser Boston) · Zbl 0474.52007 [7] Kenderov, P. S., Polygonal approximation of plane convex compacta, J. Approx. Theory, 38, 221-239 (1983) · Zbl 0521.41021 [8] McClure, D. E.; Vitale, R. A., Polygonal approximation of plane convex bodies, J. Math. Anal. Appl., 51, 326-358 (1975) · Zbl 0315.52004 [10] Weil, W., Ein approximationssatz für konvexe Körper, Manuscripta Math., 8, 335-362 (1973) · Zbl 0251.52007 [11] Shephard, G. C.; Webster, R. J., Metrics for sets of convex bodies, Mathematika, 12, 73-88 (1965) · Zbl 0144.21303 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.