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).

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 |