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The polar dual of a convex polyhedral set in hyperbolic space. (English) Zbl 0860.52007
Mich. Math. J. 42, No. 3, 479-510 (1995); Correction 43, No. 3, 619 (1996).
Let $$V$$ be a finite-dimensional vector space. A subset $$X$$ of $$V$$ is a convex polyhedral set (polyhedral cone $$C$$) if it is defined by a finite number of affine (linear) inequalities. Let $$X$$ be a convex polyhedral set in a space of constant curvature and denote by $$P(X)$$ its polar dual. A piecewise spherical cell complex, with its intrinsic metric, is “large” if there is a unique geodesic between any two points of distance less than $$\pi$$. A polyhedron of piecewise constant curvature has curvature bounded from above if and only if the link of each point is large. Let $$S_\infty(X)$$ denote the points at infinity of $$X$$. For any $$y\in S_\infty(X)$$, let $$F_y$$ be the unique face of $$C(X)$$ that contains $$y$$ in its relative interior. The point $$y$$ is called a “cusp” point of $$X$$ if $$F_y$$ is lightlike. If $$y$$ is a cusp point of $$X$$, then put $$P_y= P_{F_y}= F_y^\perp\cap P(X)$$.
The main result is: Suppose $$X$$ is a hyperbolic convex polyhedral set of dimension $$n$$. Then, (1) its polar dual $$P(X)$$ is large, and (2) if $$\gamma$$ is any closed local geodesic of length $$2\pi$$, then $$\gamma$$ must lie in the subcomplex $$P_y$$ for some cusp point $$y$$ of $$X$$.
The last sections relate the main result to the following conjecture: For a $$p$$-dimensional spherical cell $$\sigma$$, let $$a(\sigma)$$ denote the $$p$$-dimensional volume suitable normalized and let $$a^*(\sigma)= a(\sigma^*)$$ where $$\sigma^*$$ is the dual cell to $$\sigma$$. Given a finite, piecewise spherical cell complex $$K$$, consider the quantity $$\kappa(K)= 1+\sum_{\sigma^{-1}} (\dim \sigma)+1\cdot a^*(\sigma)$$, where the summation is over all cells $$\sigma$$ of $$K$$. Suppose that $$K$$ is a large piecewise spherical structure on the $$(2m-1)$$-sphere, then $$(-1)^m \kappa(K)\geq 0$$. The sign of $$\kappa(K)$$ is correct when $$K$$ is the polar dual of a hyperbolic polytope.

##### MSC:
 52B70 Polyhedral manifolds 52A55 Spherical and hyperbolic convexity 53A35 Non-Euclidean differential geometry 53C22 Geodesics in global differential geometry
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