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.

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.

Reviewer: L.A.Santaló (Buenos Aires)