View-obstruction Problem Research Articles

The view-obstruction problem for polygons

The view-obstruction problem for polygons

Minmax problems for fractional parts of real numbers

The view-obstruction problem for the n-dimensional cube with side 1 can be inter- preted as the problem of evaluating the function κ(n )= inf max0x1 min1knvkx � , where the infimum is taken over all n-tuples v1, ··· ,vn of positive integers. So the following could perhaps be called generalized view-obstruction problems: given a periodic function φ , an interval I and a set of integers S ,f ind

The View-Obstruction Problem for 5-Dimensional Cubes

The view-obstruction problem for the n-dimensional cube is equivalent to the conjecture that for any n positive integers there is a real number x such that each (here denotes the distance from y to the nearest integer). This conjecture has been previously solved for . In this paper we prove that when we can find x which gives each ; this is the first improvement over the easy result .

The View-Obstruction Problem forn-Dimensional Cubes

The view-obstruction problem for then-dimensional cube is equivalent to the conjecture that for anynpositive integersv1,…,vnthere is a real numberxsuch that each ‖vix‖⩾(n+1)−1(here ‖y‖ denotes the distance fromyto the nearest integer). This conjecture has been previously solved forn⩽4. In this paper we prove that when 2n−3 is a prime andn⩾4 we can find a real numberxwhich gives each ‖vix‖⩾1/(2n−3). In fact more than this is proved.

Contributions to a General Theory of View-Obstruction Problems, II

In view-obstruction problems, congruent copies of a closed, centrally symmetric, convex bodyC, centred at the points of the shifted lattice (12, 12, ..., 12)+Znin Rn, are expanded uniformly. The expansion factor required to touch a given subspaceLis denoted byν(C,L) and for each dimensiond, 1≤d≤n−1, the relevant expansion factors are used to determine a supremumν(C,d)=sup {ν(C,L): dimL=d,Lnot contained in a coordinate hyperplane}.Here a method for obtaining upper bounds onν(C,L) for “rational” subspacesLis given. This leads to many interesting results, e.g. it follows that the supremaν(C,d) are always attained and a general isolation result always holds. The method also applies to give simple proofs of known results for three dimensional spheres. These proofs are generalized to obtainν(B,n−2) and a Markoff type chain of related isolations for spheresBin Rnwithn≥4. In another part of the paper, the subspaces occurring in view-obstruction problems are generalized to arbitrary flats. This generalization is related to Schoenberg's problem of billiard ball motion. Several results analogous to those forν(C,L) andν(C,d) are obtained.

The view-obstruction problem for spheres in ?5

In the view-obstruction problem, congruent, closed convex bodies centred at the points\(\frac{1}{2},{\text{ }}...{\text{ , }}\frac{{\text{1}}}{{\text{2}}} + \mathbb{Z}^n \) in ℝn are expanded uniformly until they block all rays from the origin into the open positive cone. The central problem is to determine the minimal blocking size. In the case of spheres of diameter 1 and cubes of side 1 these values are known forn=2, 3 and 4. Here we show that in ℝ5, this value for the sphere of diameter 1 is\(\sqrt {41/42} \).

View-Obstruction Problems and Kronecker's Theorem

View-Obstruction Problems and Kronecker's Theorem

View-obstruction problems and Kronecker’s theorem

In this paper we show how the quantitative forms of Kronecker’s theorem in Diophantine approximations can be applied to investigate view-obstruction problems. In particular we answer a question in [Yong-Gao Chen, On a conjecture in Diophantine approximation, III, J. Number Theory 39 (1991), 91-103].

The View-Obstruction Problem for 4-Dimensional Spheres

In this paper, it has been proved that v(A) ll/\/l65. View-obstruction problems for n-dimensional geometry were formulated by T. W. Cusick [1] in 1973. Cusick [1] proved that v(2) = l/>/5. V. C. Dumir and R. J. Hans-Gill [5] proved that u(3) = 3/y/ll. Yong-gao Chen [8] gave another proof of v(3) = 3/\/2T. As for progress of view-obstruction problems for n-dimensional cubes one may refer to T. W. Cusick [1^4] and Yong-gao Chen [6^8]. It is easy to see that

A Markoff-type chain for the view-obstruction problem for spheres in ?4

In the view-obstruction problem, congruent, closed, convex bodies centred at the points $$(\frac{1}{2},\frac{1}{2},...,\frac{1}{2}) + \mathbb{Z}^n $$ in ℝ n are expanded uniformly until they block all rays from the origin into the open positive cone. The central problem is to determine the minimal blocking size. In the case of spheres of diameter 1, this value is denoted byv(n) and is known for dimensionsn=2,3. Here we show that $$v(4) = \sqrt {11/15} $$ and obtain a Markoff type chain of isolated extreme values.

The view-obstruction problem for 3-dimensional spheres

LetC be ann-dimensional sphere with diameter 1 and center at the origin inE n . The view-obstruction problem forn-dimensional spheres is to determine a constant ν(n) to be the lower bound of those α for which any half-lineL, given byx i =a i t (i=1,2,...,n) where parametert≥0 anda i (i=1,2,...,n) are positive real numbers, intersects $$\Delta (C,\alpha ) = \left\{ {\alpha C + \left( {m_1 + \frac{1}{2},m_2 + \frac{1}{2},...,m_n + \frac{1}{2}} \right):m_1 ,m_2 ,...,m_n nonnegative integers} \right\}$$ . In this paper, forn=3, the following result is proved. For α > $$1/\sqrt 2 $$ we have that any half-lineL, given byx i =a i t(i=1,2,3), intersects Δ(C,α), where parametert≥0 anda i (i=1,2,3) are positive real numbers such that |a|+|b|+|c|≠3 wheneveraa 1+ba 2+ca 3=0 for three integersa,b,c.

Contributions to a General Theory of View-Obstruction Problems

AbstractIn the original view-obstruction problem congruent closed, centrally symmetric convex bodies centred at the points of the set are expanded uniformly until they block all rays from the origin into the open positive cone. The central problem is to determine the minimal blocking size and this value is known for balls in dimensions n = 2,3 and for symmetrically placed cubes in dimensions n = 2, 3, 4In order to explain fully the distinction between rational and irrational rays in the original problem, we extend consideration to the blocking of subspaces of all dimensions. In order to appreciate the special properties of balls and cubes, we give a discussion of the convex body with respect to reflection symmetry, lower dimensional sections, and duality. We introduce topological considerations to help understand when the critical parameter of the theory is an attained maximum and we add substantially to the list of known values of this parameter. In particular, when the dimension is n = 2 our dual body considerations furnish a complete solution to the view-obstruction problem

View-obstruction problem for 3-dimensional spheres

The view-obstruction problem for ann-dimensional closed convex bodyC containing the origin in its interior has been formulated by T. W. Cusick in 1973. The problem is to determine the constantK(C) defined to be the infimum of those α>0 such that for any rayL starting from the origin and lying in the positive quadrant there are non-negative integersm1,...,mn for whichL intersects αC+(m1+1/2,...,mn+1/2). Here a conjecture of T. W. Cusick for the 3-dimensional spheres with centre 0 is proved. In fact an infinite sequence of isolated minima is obtained.

View-obstruction problems, III

Suppose C is a closed convex body in E n which contains the origin as an interior point. Define αC for each real number α ≥ 0 to be the magnification of C by the factor α and define C + ( m 1, …, m n ) for each point ( m 1, …, m n ) in E n to be the translation of C by the vector ( m 1, …, m n ). Define the point set Δ( C, α) by Δ(C, α) = {αC + (m 1 + 1 2 , …, m n + 1 2 ): m 1, …, m n nonnegative integers} . The view-obstruction problem for C is the problem of finding the constant K( C) defined to be the lower bound of those α such that any half-line L given by x i = a i t ( i = 1, 2, …, n), where the a i (1 ≤ i ≤ n) are positive real numbers and the parameter t runs through [0, ∞), intersects Δ( C, α). The paper considers the case where C is the n-dimensional cube with side 1, and in this case the constant K( C) is evaluated for n = 4. The proof in dimension 4 depends on a theorem (proved via exponential sums) concerning the existence of solutions for a certain system of simultaneous congruences. The proofs in dimensions 2 and 3 are much simpler, and for these dimensions several other proofs have previously been given. For real x, let | x| denote the distance from x to the nearest integer. A non-geometric description of our principal result is that we prove the case n = 4 of the following conjecture: For any n positive integers w 1, …, w n there is a real number x such that each | w i x| ≥ ( n + 1) −1.

View-Obstruction Problems. II

Let ${S^n}$ denote the region $0 < {x_i} < \infty (i = 1,2, \ldots ,n)$ of $n$-dimensional Euclidean space ${E^n}$. Suppose $C$ is a closed convex body in ${E^n}$ which contains the origin as an interior point. Define $\alpha C$ for each real number $\alpha \geqslant 0$ to be the magnification of $C$ by the factor $\alpha$ and define $C + ({m_1}, \ldots ,{m_n})$ for each point $({m_1}, \ldots ,{m_n})$ in ${E^n}$ to be the translation of $C$ by the vector $({m_1}, \ldots ,{m_n})$. Define the point set $\Delta (C,\alpha )$ by $\Delta (C,\alpha ) = \{ \alpha C + ({m_1} + \frac {1} {2}, \ldots ,{m_n} + \frac {1} {2}):{m_1}, \ldots ,{m_n}$ nonnegative integers}. The view-obstruction problem for $C$ is the problem of finding the constant $K(C)$ defined to be the lower bound of those $\alpha$ such that any half-line $L$ given by ${x_i} = {a_i}t(i = 1,2, \ldots ,n)$, where the ${a_i}(1 \leqslant i \leqslant n)$ are positive real numbers, and the parameter $t$ runs through $[0,\infty )$, intersects $\Delta (C,\alpha )$. The paper considers the case where $C$ is the $n$-dimensional cube with side 1, and in this case the constant $K(C)$ is known for $n \leqslant 3$. The paper gives a new proof for the case $n = 3$. Unlike earlier proofs, this one could be extended to study the cases with $n \geqslant 4$.

View-obstruction problems. II

Let S n {S^n} denote the region 0 &gt; x i &gt; ∞ ( i = 1 , 2 , … , n ) 0 &gt; {x_i} &gt; \infty (i = 1,2, \ldots ,n) of n n -dimensional Euclidean space E n {E^n} . Suppose C C is a closed convex body in E n {E^n} which contains the origin as an interior point. Define α C \alpha C for each real number α ⩾ 0 \alpha \geqslant 0 to be the magnification of C C by the factor α \alpha and define C + ( m 1 , … , m n ) C + ({m_1}, \ldots ,{m_n}) for each point ( m 1 , … , m n ) ({m_1}, \ldots ,{m_n}) in E n {E^n} to be the translation of C C by the vector ( m 1 , … , m n ) ({m_1}, \ldots ,{m_n}) . Define the point set Δ ( C , α ) \Delta (C,\alpha ) by Δ ( C , α ) = { α C + ( m 1 + 1 2 , … , m n + 1 2 ) : m 1 , … , m n \Delta (C,\alpha ) = \{ \alpha C + ({m_1} + \frac {1} {2}, \ldots ,{m_n} + \frac {1} {2}):{m_1}, \ldots ,{m_n} nonnegative integers}. The view-obstruction problem for C C is the problem of finding the constant K ( C ) K(C) defined to be the lower bound of those α \alpha such that any half-line L L given by x i = a i t ( i = 1 , 2 , … , n ) {x_i} = {a_i}t(i = 1,2, \ldots ,n) , where the a i ( 1 ⩽ i ⩽ n ) {a_i}(1 \leqslant i \leqslant n) are positive real numbers, and the parameter t t runs through [ 0 , ∞ ) [0,\infty ) , intersects Δ ( C , α ) \Delta (C,\alpha ) . The paper considers the case where C C is the n n -dimensional cube with side 1, and in this case the constant K ( C ) K(C) is known for n ⩽ 3 n \leqslant 3 . The paper gives a new proof for the case n = 3 n = 3 . Unlike earlier proofs, this one could be extended to study the cases with n ⩾ 4 n \geqslant 4 .

View-obstruction problems in n-dimensional geometry

Let S n denote the region 0 < x i < ∞ ( i = 1,2,…, n) of n-dimensional Euclidean space E n . Suppose C is a closed convex body in E n which contains the origin as an interior point. Define αC for each real number α ⩾ 0 to be the set of all ( αx 1,…, αx n ), where ( x 1,…, x n ) is a point in C. Define C + ( m 1,…, m n ) for each point ( m 1,…, m n ) of E n to be the set of all ( x 1 + m 1,…, x n + m n ), where ( x 1,…, x n ) is a point in C. Define the point set Δ( C, α) by Δ(C, α) = {αC + (m 1 + 1 2 ,…,m n + 1 2 ): m 1,…,m n non-negative integers}. The view-obstruction problem for C is the problem of finding the constant K( C) defined to be the lower bound of those α such that any half-line L given by x i = a i t ( i = 1,2,…, n), where the a i (1 ⩽ i ⩽ n) are positive real numbers and the parameter t runs through [0, ∞], intersects Δ( C, α). The simplest choices for C are the n-dimensional cube with side 1 and the n-dimensional sphere with diameter 1: let λ( n) and ν( n), respectively, denote the constant K( C) in these cases. Elementary geometry is enough to prove that λ(2) = 1 3 and ν(2) = 1 √5 . The paper uses a method partly analytic and partly combinatorial to prove that λ(3) = 1 2 . The conjecture λ(n) = (n −1) (n + 1) for each n ⩾ 2 is stated, and a connection with a certain Diophantine approximation problem is shown.

View-obstruction problems

View-obstruction problems

View-obstruction problems

View-obstruction problems