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Width and related invariants of Riemannian manifolds. (English) Zbl 0684.53036
On the geometry of differentiable manifolds, Workshop, Rome/Italy 1986, Astérisque 163-164, 93-109 (1988).
[For the entire collection see Zbl 0666.00013.]
The author poses and studies the question of how to measure the size of a metric space, in particular of a Riemannian manifold. The main notion (due to Urysohn) is that of intermediate diameters $$Diam_ k$$ for $$k=0,...,n-1$$ where n is the dimension of the space. $$Diam_ 0$$ is the usual diameter and $$Diam_ k$$ measures the $$(k+1)$$-dimensional spread of the space.
In the first part, some properties of this notion for convex and compact subsets (in particular rectangular solids) of $${\mathbb{R}}^ n$$, which follow from classical results are discussed: (i) a lemma of Lebesgue which relates $$Diam_ k$$ of a compact convex subset in $${\mathbb{R}}^ n$$ to $$Wid_ k$$, where $$Wid_ k$$ (the k-dimensional width) measures how close a subset is to a k-dimensional affine subspace. (ii) the isoperimetric inequality of Federer-Fleming which gives a sharp upper bound for $$Diam_{k-1}$$ in terms of the Hausdorff volume $$Vol_ k$$ for compact subsets of $${\mathbb{R}}^ n.$$
In the second part, the problem of getting upper bounds for the intermediate diameter for subsets of a Riemannian manifold under curvature restraints is studied. For example, the Federer-Fleming isoperimetric inequality is generalized to compact subsets of any Riemannian manifold with nonnegative Ricci curvature and $$Diam_{n-2}$$ is conjectured to be bounded from above in terms of a positive lower bound for the scalar curvature. Many of the ideas in this paper are closely related to the author’s previous work, especially his paper in J. Differ. Geom. 18, 1-147 (1983; Zbl 0515.53037).
Reviewer: M.Min-Oo

##### MSC:
 53C20 Global Riemannian geometry, including pinching