On the Number of Flats Tangent to Convex Hypersurfaces in Random Position

Abstract

Motivated by questions in real enumerative geometry (Borcea et al., in Discrete Comput Geom 35(2):287–300, 2006; Bürgisser and Lerario, in J Reine Angew Math, https://doi.org/10.1515/crelle-2018-0009, 2018; Megyesi and Sottile, in Discrete Comput Geom 33(4):617–644, 2005; Megyesi et al., in Discrete Comput Geom 30(4):543–571, 2003; Sottile and Theobald, in Trans Am Math Soc 354(12):4815–4829, 2002; Proc Am Math Soc 133(10):2835–2844, 2005; in: Goodman et al., in Surveys on discrete and computational geometry. AMS, Providence, 2008) we investigate the problem of the number of flats simultaneously tangent to several convex hypersurfaces in real projective space from a probabilistic point of view (here by “convex hypersurfaces” we mean that these hypersurfaces are boundaries of convex sets). More precisely, we say that smooth convex hypersurfaces $$X_1, \ldots , X_{d_{k,n}}\subset {\mathbb {R}}\text {P}^n$$, where $$d_{k,n}=(k+1)(n-k)$$, are in random position if each one of them is randomly translated by elements $$g_1, \ldots , g_{{d_{k,n}}}$$ sampled independently from the orthogonal group with the uniform distribution. Denoting by $$\tau _k(X_1, \ldots , X_{d_{k,n}})$$ the average number of k-dimensional projective subspaces (k-flats) which are simultaneously tangent to all the hypersurfaces we prove that $$\begin{aligned} \tau _k(X_1, \ldots , X_{d_{k,n}})={\delta }_{k,n} \cdot \prod _{i=1}^{d_{k,n}}\frac{|\Omega _k(X_i)|}{|\text {Sch}(k,n)|}, \end{aligned}$$where $${\delta }_{k,n}$$ is the expected degree from [6] (the average number of k-flats incident to $$d_{k,n}$$ many random $$(n-k-1)$$-flats), $$|\text {Sch}(k,n)|$$ is the volume of the Special Schubert variety of k-flats meeting a fixed $$(n-k-1)$$-flat (computed in [6]) and $$|\Omega _k(X)|$$ is the volume of the manifold $$\Omega _k(X)\subset \mathbb {G}(k,n)$$ of all k-flats tangent to X. We give a formula for the evaluation of $$|\Omega _k(X)|$$ in terms of some curvature integral of the embedding $$X\hookrightarrow {\mathbb {R}}\text {P}^n$$ and we relate it with the classical notion of intrinsic volumes of a convex set: $$\begin{aligned} \frac{|\Omega _{k}(\partial C)|}{|\text {Sch}(k, n)|}=4V_{n-k-1}(C),\quad k=0, \ldots , n-1. \end{aligned}$$As a consequence we prove the universal upper bound: $$\begin{aligned} \tau _k(X_1, \ldots , X_{d_{k,n}})\le {\delta }_{k, n}\cdot 4^{d_{k,n}}. \end{aligned}$$Since the right hand side of this upper bound does not depend on the specific choice of the convex hypersurfaces, this is especially interesting because already in the case $$k=1, n=3$$ for every $$m>0$$ we can provide examples of smooth convex hypersurfaces $$X_1, \ldots , X_4$$ such that the intersection $$\Omega _1(X_1)\cap \cdots \cap \Omega _1(X_4)\subset \mathbb {G}(1,3)$$ is transverse and consists of at least m lines. Finally, we present analogous results for semialgebraic hypersurfaces (not necessarily convex) satisfying some nondegeneracy assumptions.