On the number of flats tangent to convex hypersurfaces in random position

Abstract

We investigate the problem of the number of flats simultaneously tangent to several convex hypersurfaces in real projective space from a random point of view. More precisely, we say that smooth convex hypersurfaces $X_1, \ldots, X_{d_{k,n}}\subset \mathbb{R}\textrm{P}^n$, 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 and uniformly from the Orthogonal group; we denote by $\tau_k(X_1, \ldots, X_{d_{k,n}})$ the average number of $k$-dimensional projective subspaces (flats) which are simultaneously tangent to all the hypersurfaces. We prove that $$\tau_k(X_1, \ldots, X_{d_{k,n}})={\delta}_{k,n}\cdot\prod_{i=1}^{d_{k,n}}\frac{|\Omega_k(X_i)|}{|\textrm{Sch}(k,n)|},$$ where ${\delta}_{k,n}$ is the expected degree (the average number of $k$-flats incident to $d_{k,n}$ many random $(n-k-1)$-flats), $|\textrm{Sch}(k,n)|$ is the volume of the Special Schubert variety of $k$-flats meeting a $(n-k-1)$-flat and $|\Omega_k(X)|$ is the volume of the manifold of all $k$-flats tangent to $X$. We give a formula for the evaluation of $|\Omega_k(X)|$ in term of some curvature integral of the embedding $X\hookrightarrow \mathbb{R}\textrm{P}^n$ and we relate it with the notion of intrinsic volumes of a convex set: $$\frac{|\Omega_{k}(\partial C)|}{|\textrm{Sch}(k, n)|}=4|V_{n-k-1}(C)|,\quad k=0,\ldots,n-1.$$ We prove the upper bound: $$\tau_k(X_1,\ldots,X_{d_{k,n}})\leq{\delta}_{k, n}\cdot 4^{d_{k,n}}.$$ In the case $k=1,n=3$ for every $m>0$ we 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. We also present analogous results for semialgebraic hypersurfaces satisfying some nondegeneracy assumption.