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Register a new userA Directional Curvature Formula for Convex Bodies in R^n
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F. F. Pereira
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For a compact convex set F in R^n, with the origin in its interior, we present a formula to compute the curvature at a fixed point on its boundary, in the direction of any tangent vector. This formula is equivalent to the existing ones, but it is easier to apply.
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G. Fejes Toth posed the following problem: Determine the infimum of the densities of the lattices of closed balls in $bR^n$ such that each affine $k$-subspace $(0 le k le n-1)$ of $bR^n$ intersects some ball of the lattice. We give a lower estimate for any $n,k$ like above. If, in the problem posed by G. Fejes Toth, we replace the ball $B^n$ by a (centrally symmetric) convex body $Ksubset bR^n$, we may ask for the infimum of all above infima of densities of lattices of translates of $K$ with the above property, when $K$ ranges over all (centrally symmetric) convex bodies in $bR^n$. For these quantities we give lower estimates as well, which are sharp, or almost sharp, for certain classes of convex bodies $K$. For $k=n-1$ we give an upper estimate for the supremum of all above infima of densities, $K$ also ranging as above (i.e., a minimax problem). For $n=2$ our estimate is rather close to the conjecturable maximum. We point out the connection of the above questions to the following problem: Find the largest radius of a cylinder, with base an $(n-1)$-ball, that can be fitted into any lattice packing of balls (actually, here balls can be replaced by some convex bodies $K subset bR^n$, the axis of the cylinder may be $k$-dimensional and its basis has to be chosen suitably). Among others we complete the proof of a theorem of I. Hortobagyi from 1971. Our proofs for the lower estimates of densities for balls, and for the cylinder problem, follow quite closely a paper of J. Horvath from 1970. This paper is also an addendum to a paper of the first named author from 1978 in the sense that to some arguments given there not in a detailed manner, we give here for all of these complete proofs.
We define a set inner product to be a function on pairs of convex bodies which is symmetric, Minkowski linear in each dimension, positive definite, and satisfies the natural analogue of the Cauchy-Schwartz inequality (which is not implied by the other conditions). We show that any set inner product can be embedded into an inner product space on the associated support functions, thereby extending fundamental results of Hormander and Radstrom. The set inner product provides a geometry on the space of convex bodies. We explore some of the properties of that geometry, and discuss an application of these ideas to the reconstruction of ancestral ecological niches in evolutionary biology.
We study a long standing open problem by Ulam, which is whether the Euclidean ball is the unique body of uniform density which will float in equilibrium in any direction. We answer this problem in the class of origin symmetric n-dimensional convex bodies whose relative density to water is 1/2. For n=3, this result is due to Falconer.
In this paper we investigate the existence of closed billiard trajectories in not necessarily smooth convex bodies. In particular, we show that if a body $Ksubset mathbb{R}^d$ has the property that the tangent cone of every non-smooth point $qin partial K$ is acute (in a certain sense) then there is a closed billiard trajectory in $K$.
Barker and Larman asked the following. Let $K subset {Bbb{R}}^d$ be a convex body, whose interior contains a given convex body $K subset {Bbb{R}}^d$, and let, for all supporting hyperplanes $H$ of $K$, the $(d-1)$-volumes of the intersections $K cap H$ be given. Is $K$ then uniquely determined? Yaskin and Zhang asked the analogous question when, for all supporting hyperplanes $H$ of $K$, the $d$-volumes of the caps cut off from $K$ by $H$ are given. We give local positive answers to both of these questions, for small $C^2$-perturbations of $K$, provided the boundary of $K$ is $C^2_+$. In both cases, $(d-1)$-volumes or $d$-volumes can be replaced by $k$-dimensional quermassintegrals for $1 le k le d-1$ or for $1 le k le d$, respectively. Moreover, in the first case we can admit, rather than hyperplane sections, sections by $l$-dimensional affine planes, where $1 le k le l le d-1$. In fact, here not all $l$-dimensional affine subspaces are needed, but only a small subset of them (actually, a $(d-1)$-manifold), for unique local determination of $K$.
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