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Register a new userOn the Circle Covering Theorem by A. W. Goodman and R. E. Goodman
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In 1945, A. W. Goodman and R. E. Goodman proved the following conjecture by P. ErdH{o}s: Given a family of (round) disks of radii $r_1$, $ldots$, $r_n$ in the plane it is always possible to cover them by a disk of radius $R = sum r_i$, provided they cannot be separated into two subfamilies by a straight line disjoint from the disks. In this note we show that essentially the same idea may work for different analogues and generalizations of their result. In particular, we prove the following: Given a family of positive homothetic copies of a fixed convex body $K subset mathbb{R}^d$ with homothety coefficients $tau_1, ldots, tau_n > 0$ it is always possible to cover them by a translate of $frac{d+1}{2}left(sum tau_iright)K$, provided they cannot be separated into two subfamilies by a hyperplane disjoint from the homothets.
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Goodmans theorem (1976) states that intuitionistic finite-type arithmetic plus the axiom of choice plus the axiom of relativized dependent choice is conservative over Heyting arithmetic. The same result applies to the extensional variant. This is due to Beeson (1979). In this paper we modify Goodman realizability (1978) and provide a new proof of the extensional case.
The paper is devoted to coverings by translative homothets and illuminations of convex bodies. For a given positive number $alpha$ and a convex body $B$, $g_{alpha}(B)$ is the infimum of $alpha$-powers of finitely many homothety coefficients less than 1 such that there is a covering of $B$ by translative homothets with these coefficients. $h_{alpha}(B)$ is the minimal number of directions such that the boundary of $B$ can be illuminated by this number of directions except for a subset whose Hausdorff dimension is less than $alpha$. In this paper, we prove that $g_{alpha}(B)leq h_{alpha}(B)$, find upper and lower bounds for both numbers, and discuss several general conjectures. In particular, we show that $h_{alpha} (B) > 2^{d-alpha}$ for almost all $alpha$ and $d$ when $B$ is the $d$-dimensional cube, thus disproving the conjecture from Research Problems in Discrete Geometry by Brass, Moser, and Pach.
We define new natural variants of the notions of weighted covering and separation numbers and discuss them in detail. We prove a strong duality relation between weighted covering and separation numbers and prove a few relations between the classical and weighted covering numbers, some of which hold true without convexity assumptions and for general metric spaces. As a consequence, together with some volume bounds that we discuss, we provide a bound for the famous Levi-Hadwiger problem concerning covering a convex body by homothetic slightly smaller copies of itself, in the case of centrally symmetric convex bodies, which is qualitatively the same as the best currently known bound. We also introduce the weighted notion of the Levi-Hadwiger covering problem, and settle the centrally-symmetric case, thus also confirm Nasz{o}dis equivalent fractional illumination conjecture in the case of centrally symmetric convex bodies (including the characterization of the equality case, which was unknown so far).
Given a convex disk $K$ and a positive integer $k$, let $vartheta_T^k(K)$ and $vartheta_L^k(K)$ denote the $k$-fold translative covering density and the $k$-fold lattice covering density of $K$, respectively. Let $T$ be a triangle. In a very recent paper, K. Sriamorn proved that $vartheta_L^k(T)=frac{2k+1}{2}$. In this paper, we will show that $vartheta_T^k(T)=vartheta_L^k(T)$.
Let $C$ be the unit circle in $mathbb{R}^2$. We can view $C$ as a plane graph whose vertices are all the points on $C$, and the distance between any two points on $C$ is the length of the smaller arc between them. We consider a graph augmentation problem on $C$, where we want to place $kgeq 1$ emph{shortcuts} on $C$ such that the diameter of the resulting graph is minimized. We analyze for each $k$ with $1leq kleq 7$ what the optimal set of shortcuts is. Interestingly, the minimum diameter one can obtain is not a strictly decreasing function of~$k$. For example, with seven shortcuts one cannot obtain a smaller diameter than with six shortcuts. Finally, we prove that the optimal diameter is $2 + Theta(1/k^{frac{2}{3}})$ for any~$k$.
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