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.