Large curvature on typical convex surfaces (Q2903482)

Victor Klee proved (1959) that most convex bodies in \(\mathbb R^d\) are strictly convex and smooth (i.e., the boundary -- the outer surface -- is of class \(C^1\)). The term `most' is to mean all, except those belonging to a set of first Baire category. On the other hand, Peter Gruber demonstrated (1977) that most convex bodies, although they may be smooth, are not of class \(C^2\) [\textit{P. M. Gruber}, Math. Ann. 229, 259--266 (1977; Zbl 0342.52009)]. The present paper is concerned with estimates of lower and upper curvatures of convex surfaces at a point, in particular how large can a lower curvature get in every tangent direction.NEWLINENEWLINEGiven a smooth strictly convex body \(K,\) consider a point on its boundary, \(x \in \text{bd}\, K,\) and let \(\tau \) be a unit tangent vector at \(x,\) with \(\nu\) being a unit normal at \(x.\) The intersection of the surface and the 2-dimensional half-plane \(H = \{ \alpha \tau + \beta \nu : \, \alpha > 0, \; \beta \in \mathbb R\}\) is a curve on the surface emanating from \(x.\) If \(z\) is any other point on this curve then there is a unique circle with center on the normal line \(x + \mathbb R \nu\) containing both \(x\) and \(z.\) Denote its radius by \(r_z\) and define lower and upper curvature radius respectively by NEWLINE\[NEWLINE \rho_i^{\tau}(x) = \liminf_{z \to x} r_z,\quad \rho_s^{\tau}(x) = \limsup_{z\to x} r_z. NEWLINE\]NEWLINE The cross reciprocals \(\gamma_i^{\tau}(x) = 1/\rho_s^{\tau}(x)\) and \(\gamma_s^{\tau}(x) = 1/\rho_i^{\tau}(x)\) define respectively the upper and the lower curvature of \(\text{bd}\, K\) at \(x\) in the direction of \(\tau\). When these two agree, they define the curvature \(\gamma^{\tau}(x)\) at \(x\) in the indicated direction. The second author has shown [\textit{T. Zamfirescu}, Math. Z. 174, 135--139 (1980; Zbl 0423.53003)] that for most convex bodies (which are smooth and strictly convex) at each \(x \in \text{bd}\, K\) and any tangent direction \(\tau\) at \(x,\) one has \(\gamma_i^{\tau}(x) = 0\) or \(\gamma_s^{\tau}(x) = \infty\). Combined with A. D. Alexandrov's theorem about the existence of a finite curvature a.e. in all tangent directions, one concludes that on most convex surfaces it holds \(\gamma^{\tau}(x) = 0\) almost everywhere in all tangent directions \(\tau\) at \(x\). These results preclude generic existence of points where a finite non-zero curvature in some tangent direction exists.NEWLINENEWLINEOne related question that the authors raise and try to answer is the following: Do there exist, on most convex surfaces, points with arbitrary large lower curvature in every tangent direction? Towards that goal they show that for most convex bodies \(K\) and any number \(r \in \mathbb R\), there are densely many points \(x \in \text{bd}\, K\) such that \(\gamma_i^{\tau}(x) > r\) and \(\gamma_s^{\tau}(x) = \infty\) in all tangent directions \(\tau.\) Moreover, they prove that for nearly all convex bodies \(K\) and any \(r \in \mathbb R\) there is a point \(x \in \text{bd}\, K\) with \(\gamma_i^{\tau}(x) > r\) in all tangent directions at \(x\). (here `nearly all' means all but members of some countable union of porous sets, these terms defined in a particular way). For a smooth strictly convex body \(K\) and any \(x \in \text{bd}\, K\) its opposite point \(x^{*}\) is defined as a unique point on the boundary such that hyperplanes tangents to \(\text{bd}\, K\) at \(x\) and \(x^{*}\) are parallel. Then the authors prove that on most convex surfaces there exists no point \(x\) and no tangent direction \(\tau\) such that \(\gamma_s^{\tau}(x) < \infty\) and \(\gamma_s^{\tau}(x^{*}) < \infty\).