Hyperplans poissoniens et compacts de Steiner

1974 ◽  
Vol 6 (3) ◽  
pp. 563-579 ◽  
Author(s):  
G. Matheron
Keyword(s):  
Stationary Process ◽  
Convex Set ◽  
Compact Convex ◽  
Minkowski Sum ◽  
Isotropic Case ◽  
Geometrical Properties ◽  
Steiner Formula ◽  
Compact Convex Set ◽  
Finite Sums ◽  
Linear Variety

A compact convex set in RN is Steiner if it is a finite Minkowski sum of line segments, or a limit of such finite sums, and then satisfies an extension of the Steiner formula. With each Poisson hyperplane stationary process A is uniquely associated a Steiner set M, and for any linear variety V, the Steiner set associated with is the projection of M on V. The density of the order k network Ak (i.e., the set of the intersections of k hyperplanes belonging to A) is linked with simple geometrical properties of M. In the isotropic case, the expression of the covariance measures associated with Ak is derived and compared with the analogous results obtained for (N — k)-dimensional Poisson flats.

Hyperplans poissoniens et compacts de Steiner

1974 ◽  
Vol 6 (03) ◽  
pp. 563-579 ◽  
Author(s):  
G. Matheron
Keyword(s):  
Stationary Process ◽  
Convex Set ◽  
Compact Convex ◽  
Minkowski Sum ◽  
Isotropic Case ◽  
Geometrical Properties ◽  
Steiner Formula ◽  
Compact Convex Set ◽  
Finite Sums ◽  
Linear Variety

A compact convex set in RN is Steiner if it is a finite Minkowski sum of line segments, or a limit of such finite sums, and then satisfies an extension of the Steiner formula. With each Poisson hyperplane stationary process A is uniquely associated a Steiner set M, and for any linear variety V, the Steiner set associated with is the projection of M on V. The density of the order k network Ak (i.e., the set of the intersections of k hyperplanes belonging to A) is linked with simple geometrical properties of M. In the isotropic case, the expression of the covariance measures associated with Ak is derived and compared with the analogous results obtained for (N — k)-dimensional Poisson flats.

Un théorème d'unicité pour les hyperplans poissoniens

1974 ◽  
Vol 11 (01) ◽  
pp. 184-189 ◽  
Author(s):  
G. Matheron
Keyword(s):  
Poisson Point Process ◽  
Convex Set ◽  
Compact Convex ◽  
Minkowski Sum ◽  
Line Segments ◽  
Supporting Function ◽  
Compact Convex Set ◽  
Finite Sums ◽  
Tangential Cone ◽  
Straight Lines

A stationary Poisson process of hyperplanes in Rn is characterized (up to an equivalence) by the function θ such that θ(s) is the density of the Poisson point process induced on the straight lines with direction s. The set of these functions θ is a convex cone ℛ1, a basis of which is a simplex Θ, and a given function θ belongs to ℛ1 if and only if it is the supporting function of a symmetrical compact convex set which is a finite Minkowski sum of line segments or the limit of such finite sums. Another application is given concerning the tangential cone at h = 0 of a coveriance function.

Un théorème d'unicité pour les hyperplans poissoniens

1974 ◽  
Vol 11 (1) ◽  
pp. 184-189 ◽  
Author(s):  
G. Matheron
Keyword(s):  
Poisson Point Process ◽  
Convex Set ◽  
Compact Convex ◽  
Minkowski Sum ◽  
Line Segments ◽  
Supporting Function ◽  
Compact Convex Set ◽  
Finite Sums ◽  
Tangential Cone ◽  
Straight Lines

A stationary Poisson process of hyperplanes in Rn is characterized (up to an equivalence) by the function θ such that θ(s) is the density of the Poisson point process induced on the straight lines with direction s. The set of these functions θ is a convex cone ℛ1, a basis of which is a simplex Θ, and a given function θ belongs to ℛ1 if and only if it is the supporting function of a symmetrical compact convex set which is a finite Minkowski sum of line segments or the limit of such finite sums. Another application is given concerning the tangential cone at h = 0 of a coveriance function.

scholarly journals Diameter-preserving linear bijections of function spaces

2001 ◽  
Vol 70 (3) ◽  
pp. 323-336 ◽  
Author(s):  
T. S. S. R. K. Rao ◽  
A. K. Roy
Keyword(s):  
Maximal Ideal ◽  
Convex Set ◽  
Compact Convex ◽  
Extreme Points ◽  
Continuous Functions ◽  
Ideal Space ◽  
Function Algebras ◽  
Compact Convex Set ◽  
Vector Valued Functions ◽  
Vector Valued

AbstractIn this paper we give a complete description of diameter-preserving linear bijections on the space of affine continuous functions on a compact convex set whose extreme points are split faces. We also give a description of such maps on function algebras considered on their maximal ideal space. We formulate and prove similar results for spaces of vector-valued functions.

Densities for stationary random sets and point processes

1984 ◽  
Vol 16 (02) ◽  
pp. 324-346 ◽  
Author(s):  
Wolfgang Weil ◽  
John A. Wieacker
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Convex Set ◽  
Compact Convex ◽  
Random Sets ◽  
Intensity Measure ◽  
Additive Functionals ◽  
Compact Convex Set ◽  
Classical Integral

For certain stationary random setsX, densitiesDφ(X) of additive functionalsφare defined and formulas forare derived whenKis a compact convex set in. In particular, for the quermassintegrals and motioninvariantX, these formulas are in analogy with classical integral geometric formulas. The case whereXis the union set of a Poisson processYof convex particles is considered separately. Here, formulas involving the intensity measure ofYare obtained.

Densities for stationary random sets and point processes

1984 ◽  
Vol 16 (2) ◽  
pp. 324-346 ◽  
Author(s):  
Wolfgang Weil ◽  
John A. Wieacker
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Convex Set ◽  
Compact Convex ◽  
Random Sets ◽  
Intensity Measure ◽  
Additive Functionals ◽  
Compact Convex Set ◽  
Classical Integral

For certain stationary random sets X, densities Dφ (X) of additive functionals φ are defined and formulas for are derived when K is a compact convex set in . In particular, for the quermassintegrals and motioninvariant X, these formulas are in analogy with classical integral geometric formulas. The case where X is the union set of a Poisson process Y of convex particles is considered separately. Here, formulas involving the intensity measure of Y are obtained.

Some Remarks on Translative Coverings of Convex Domains by Strips

1984 ◽  
Vol 27 (2) ◽  
pp. 233-237 ◽  
Author(s):  
H. Groemer
Keyword(s):  
Euclidean Plane ◽  
Convex Set ◽  
Compact Convex ◽  
Convex Domains ◽  
Minimal Width ◽  
Compact Convex Set

AbstractIn the euclidean plane let K be a compact convex set and Sl, S2,… strips of respective widths wl, w2,… Some conditions on Σ wi are given that imply that K can be covered by translates of the strips Si. These conditions involve the perimeter, the diameter, or the minimal width of K and yield improvements of previously known results.

Rates of convergence for random approximations of convex sets

1996 ◽  
Vol 28 (02) ◽  
pp. 384-393 ◽  
Author(s):  
Lutz Dümbgen ◽  
Günther Walther
Keyword(s):  
Convex Subset ◽  
Hausdorff Distance ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Almost Sure Convergence ◽  
Rates Of Convergence ◽  
Random Sets ◽  
Compact Convex Set

The Hausdorff distance between a compact convex set K ⊂ ℝd and random sets is studied. Basic inequalities are derived for the case of being a convex subset of K. If applied to special sequences of such random sets, these inequalities yield rates of almost sure convergence. With the help of duality considerations these results are extended to the case of being the intersection of a random family of halfspaces containing K.

Exact distributions for shapes of random triangles in convex sets

1985 ◽  
Vol 17 (02) ◽  
pp. 308-329 ◽  
Author(s):  
D. G. Kendall
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Direct Proof ◽  
Circular Disk ◽  
Compact Convex Set ◽  
Exact Evaluation ◽  
Exact Distributions ◽  
New Formula ◽  
Relevant Section

The paper starts with a simple direct proof that . A new formula is given for the shape-density for a triangle whose vertices are i.i.d.-uniform in a compact convex set K, and an exact evaluation of that shape-density is obtained when K is a circular disk. An (x, y)-diagram for an auxiliary shape-density is then introduced. When K = circular disk, it is shown that is virtually constant over a substantial region adjacent to the relevant section of the collinearity locus, large enough to contain the work-space for most collinearity studies, and particularly appropriate when the ‘strip’ method is used to assess near-collinearity.

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