## Hard and soft packing radius theorems.(English)Zbl 0846.53042

For a compact metric space $$X$$, let $$\text{pack}_q X$$ denote the $$q$$th packing radius of $$X$$, that is $2 \text{pack}_qX = \max_{(x_1, \dots, x_q) \in X^q} \min_{i < j} \text{dist} (x_i, x_j),$ where the maximum is taken over all configurations of $$q$$ points in $$X$$, and the minimum is the minimum of all pairwise distances in such a configuration. Now assume that $$X$$ is an $$n$$-dimensional Alexandrov space with curvature $$\geq 1$$. Then $$\text{pack}_qX \leq \text{pack}_q S^n_1$$ for all integers $$q \geq 2$$, where $$S^n_1$$ is the unit sphere. In particular, this inequality hold for any closed Riemannian $$n$$-manifold with sectional curvature $$\geq 1$$ (or Ricci curvature $$\geq n - 1)$$. The authors prove the following geometric (resp. topological) join theorem:
If $$\text{pack}_qX = \text{pack}_q S^n_1$$ (resp. $$\text{pack}_q X > {\pi \over 4})$$ for some $$2 \leq q \leq n + 2$$, then $$X$$ is isometric (resp. homeomorphic) to the $$(q - 1)$$-fold spherical suspension $$\sum_1^{q - 1} E \equiv S^{q - 2}_1* E$$ [resp. a $$(q - 1)$$-fold topological suspension $$\sum^{q - 1} E \cong S^{q - 2}* E]$$ of an $$(n - q + 1)$$-dimensional Alexandrov space $$E$$ with curve $$E \geq 1$$.
The key tool in the proof of the geometric version of the theorem is a rigidity distance comparison theorem [see K. Grove and S. Markvorsen, J. Am. Math. Soc. 8, No. 1, 1-28 (1995; Zbl 0829.53033) appendix], whereas the critical point theory for distance functions in Alexandrov spaces [see Yu. Burago, M. Gromov and G. Perelman, Russ. Math. Surv. 47, No. 2, 1-58 (1992); translation from Usp. Mat. Nauk 47, No. 2(284), 3-51 (1992; Zbl 0802.52018)] is used to prove the topological version.
Also, the following packing radius sphere theorem is proved: If $$n \geq 3$$, then any closed Riemannian $$n$$-manifold $$M$$ with sectional curvature $$\geq 1$$ and $$\text{pack}_{n - 1} M > {\pi \over 4}$$ is diffeomorphic to $$S^n$$. The proof is based on ideas from Alexandrov geometry, namely the global Riemannian problem associated with the theorem is changed to a local problem in Alexandrov geometry. This result generalizes differentiable sphere theorems of Y. Otsu, K. Shiohama and T. Yamaguchi [Invent. Math. 98, No. 2, 219-228 (1989; Zbl 0688.53016)], C. Plaut [Spaces of Wald curvature bounded from below, J. Geom. Anal., to appear), K. Shiohama and T. Yamaguchi [in Geometry of manifolds, Coll. Pap. 35th Symp. Differ. Geom., Matsumoto/Japan 1988, Perspect. Math. 8, 345-350 (1989; Zbl 0697.53041)], F. H. Wilhelm jun. [Invent. Math. 107, No. 3, 653-668 (1992; Zbl 0739.53036); Indiana Univ. Math. J. 41, No. 1, 1119-1142 (1992; Zbl 0771.53028)], J.-Y. Wu [in Proc. Symp. Pure Math. 54, Part 3, 685-692 (1993; Zbl 0804.53063)], since the hypotheses of these results imply that $$M$$ is close to $$S^n_1$$ in the Gromov-Hausdorff topology and consequently that $$\text{pack}_{n - 1} M > {\pi \over 4}$$. Two corollaries of the authors’ version of the differentiable sphere theorem are pointed out.

### MSC:

 53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov) 53C20 Global Riemannian geometry, including pinching

### Keywords:

packing radius; Alexandrov space; sphere theorem
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