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Register a new userPairs of convex bodies in $S^d$, ${Bbb R}^d$ and $H^d$, with symmetric intersections of their congruent copies
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High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical and hyperbolic planes. If in any of these planes, or in ${Bbb R}^2$, there is a pair of closed convex sets with interior points, and the intersections of any congruent copies of these sets are centrally symmetric, then, under some mild hypotheses, our sets are congruent circles, or, for ${Bbb R}^2$, two parallel strips. We prove the analogue of this statement, for $S^d$, ${Bbb R}^d$, $H^d$, if we suppose $C^2_+$: again, our sets are congruent balls. In $S^2$, ${Bbb R}^2$ and $H^2$ we investigate a variant of this question: supposing that the numbers of connected components of the boundaries of both sets are finite, we exactly describe all pairs of such closed convex sets, with interior points, whose any congruent copies have an intersection with axial symmetry (there are 1, 5 or 9 cases, respectively).
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In this paper, we study the following problem: Let $Dgeq 2$ and let $Esubset mathbb R^D$ be finite satisfying certain conditions. Suppose that we are given a map $phi:Eto mathbb R^D$ with $phi$ a small distortion on $E$. How can one decide whether $phi$ extends to a smooth small distortion $Phi:mathbb R^Dto mathbb R^D$ which agrees with $phi$ on $E$. We also ask how to decide if in addition $Phi$ can be approximated well by certain rigid and non-rigid motions from $mathbb R^Dto mathbb R^D$. Since $E$ is a finite set, this question is basic to interpolation and alignment of data in $mathbb R^D$.
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.
Let $Dgeq 2$, $Ssubset mathbb R^D$ be finite and let $phi:Sto mathbb R^D$ with $phi$ a small distortion on $S$. We solve the Whitney extension-interpolation-alignment problem of how to understand when $phi$ can be extended to a function $Phi:mathbb R^Dto mathbb R^D$ which is a smooth small distortion on $mathbb R^D$. Our main results are in addition to Whitney extensions, results on interpolation and alignment of data in $mathbb R^D$ and complement those of [14,15,20].
A set $cal P$ of $n$ points in $R^d$ is separated if all distances of distinct points are at least~$1$. Then we may ask how many of these distances, with multiplicity, lie in an interval $[t, t + 1]$. The authors and J. Spencer proved that the maximum is $(n^2/2)(1 - 1/d) + O(1)$. The authors showed that for $d = 2$ and $cal P$ separated, the maximal number of distances, with multiplicity, in the union of $k$ unit intervals is $(n^2/2)$ $(1 - 1/(k + 1) + o(1))$. (In these papers the unit intervals could be replaced by intervals of length $text{const}_dcdot n^{1/d}$.) In this paper we show that for $k = 2$, and for any $n$, this maximal number is $(n^2/2)(1 - 1/m_{d - 1} + o(1))$, where $m_{d - 1}$ is the maximal size of a two-distance set in $R^{d - 1}$. (The value of $m_{d - 1}$ is known for $d - 1 leq 8$, and for each $d$ it lies in $left[left({datop 2}right), left({d + 1atop 2}right)right]$. For $d eq 4,5$ we can replace unit intervals by intervals of length $text{const}_d cdot n^{1/d}$, and the maximum is the respective Turan number, for $n geq n(d)$.) We also investigate a variant of this question, namely with $k$ intervals of the form $[t, t(1 + varepsilon)]$, for $varepsilon < varepsilon (d, k)$, and for $n > n(d, k)$. Here the maximal number of distances, with multiplicity, in the union of $k$ such intervals is the Turan number $T(n, (d + 1)^k + 1)$. Several of these results were announced earlier by Makai-Pach-Spencer.
Let $Omega$ be a bounded closed convex set in ${mathbb R}^d$ with non-empty interior, and let ${cal C}_r(Omega)$ be the class of convex functions on $Omega$ with $L^r$-norm bounded by $1$. We obtain sharp estimates of the $epsilon$-entropy of ${cal C}_r(Omega)$ under $L^p(Omega)$ metrics, $1le p<rle infty$. In particular, the results imply that the universal lower bound $epsilon^{-d/2}$ is also an upper bound for all $d$-polytopes, and the universal upper bound of $epsilon^{-frac{(d-1)}{2}cdot frac{pr}{r-p}}$ for $p>frac{dr}{d+(d-1)r}$ is attained by the closed unit ball. While a general convex body can be approximated by inscribed polytopes, the entropy rate does not carry over to the limiting body. Our results have applications to questions concerning rates of convergence of nonparametric estimators of high-dimensional shape-constrained functions.
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