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Papers by Fernando Affentranger

Research paper thumbnail of Remarks on the note "Generalization of a formula of C. Buchta about the convex hull of random points

Research paper thumbnail of Generalization of a formula of C. Buchta about the convex hull of random points

Research paper thumbnail of Random approximation of convex bodies

Publicacions Matematiques, 1992

Research paper thumbnail of On the Convex Hull of Uniform Randowm Points in a Simple d-polytope

Discrete & Computational Geometry, 1991

Research paper thumbnail of Aproximación aleatoria de cuerpos convexos

Publicacions Matemàtiques, 1992

APROXIMACIÓN ALEATORIA DE CUERPOS CONVEXOS* FERNANDO AFFENTRANGER Problems related te the random ... more APROXIMACIÓN ALEATORIA DE CUERPOS CONVEXOS* FERNANDO AFFENTRANGER Problems related te the random approximation of convex bodies fall into the field of integral geometry and geometric probabilities. The aim of this paper is te give a survey of known results about the stochastic model that has received special attention in the literature and that can be described as follows : Let K be a d-dimensional convex body in Euclidean space Rd, d >_ 2. Denote by H the convex hull of n independent random points X 1,. .. , X distributed identically and uniformly in the interior of K. lf cp is a random variable en d-dimensional polytopes in Rd, we define the random variable cp, by ,pn = W(conv{X1,. .. ,Xn}), *Extended version of an invited talk,

Research paper thumbnail of Pairs of non-intersecting random flats

Probability Theory and Related Fields, 1988

Research paper thumbnail of Random circles in the d-dimensional unit ball

Journal of Applied Probability, 1989

This note gives the solution of the following problem concerning geometric probabilities. What is... more This note gives the solution of the following problem concerning geometric probabilities. What is the probability p(Bd; 2) that the circumference determined by three points P, P1 and P2 chosen independently and uniformly at random in the interior of a d-dimensional unit ball Bd in Euclidean space Ed (d ≧ 2) is entirely contained in Bd? From our result we conclude that p(Bd; 2) →π /(3√3) as d →∞.

Research paper thumbnail of Random projections of regular simplices

Discrete & Computational Geometry, 1992

Precise asymptotic formulae are obtained for the expected number of k-faces of the orthogonal pro... more Precise asymptotic formulae are obtained for the expected number of k-faces of the orthogonal projection of a regular n-simplex in n-space onto a randomly chosen isotropic subspace of fixed dimension or codimension, as the dimension n tends to infinity.

Research paper thumbnail of On the convex hull of uniform random points in a simpled-polytope

Discrete & Computational Geometry, 1991

Denote the expected number of facets and vertices and the expected volume of the convex hull Pn o... more Denote the expected number of facets and vertices and the expected volume of the convex hull Pn of n random points, selected independently and uniformly from the interior of a simple d-polytope by En(f), E.(v), and E~(V), respectively. In this note we determine the sharp constants of the asymptotic expansion of En(f), E.(v), and En(V), as n tends to infinity. Further, some results concerning the expected shape of P~ are given.

Research paper thumbnail of Random spheres in a convex body

Archiv der Mathematik, 1990

Research paper thumbnail of The Convex Hull of Random Balls

Advances in Applied Probability, 1993

While the convex hull of n d-dimensional balls is not a polytope, it does have an underlying comb... more While the convex hull of n d-dimensional balls is not a polytope, it does have an underlying combinatorial structure similar to that of a polytope. In the worst case, its combinatorial complexity can be of order Ω(n[d/2]). The thrust of this work is to show that its complexity is typically much smaller, and that it can therefore be constructed more quickly on average than in the worst case. To this end, four models of the random d-ball are developed, and the expected combinatorial complexity of the convex hull of n independent random d-balls is investigated for each. As n grows without bound, these expectations are O(1), O(n(d–1)/(d+4)), O(1) (for d = 2 only), and O(n(d–1)/(d+3)).

Research paper thumbnail of The expected volume of a random polytope in a ball*

Journal of Microscopy, 1988

For any convex body K in d‐dimensional Euclidean space Ed(d≥2) and for integers n and i, n ≥ d + ... more For any convex body K in d‐dimensional Euclidean space Ed(d≥2) and for integers n and i, n ≥ d + 1,1 ≤ i ≤ n, let V(d)n‐ii(K) be the expected volume of the convex hull Hn‐i, i of n independent random points, of which n‐i are uniformly distributed in the interior, the other i on the boundary of K. We develop an integral formula for V(d)n‐i, i(K) for the case that K is a d‐dimensional unit ball by considering an adequate decomposition of V(d)n‐i, i into d‐dimensional simplices. To solve the important case i = 0, that is the case in which all points are chosen at random from the interior of Bd, we require in addition Crofton's theorem on mean values. We illustrate the usefulness of our results by treating some special cases and by giving numerical values for the planar and the three‐dimensional cases.

Research paper thumbnail of Remarks on the note "Generalization of a formula of C. Buchta about the convex hull of random points

Research paper thumbnail of Generalization of a formula of C. Buchta about the convex hull of random points

Research paper thumbnail of Random approximation of convex bodies

Publicacions Matematiques, 1992

Research paper thumbnail of On the Convex Hull of Uniform Randowm Points in a Simple d-polytope

Discrete & Computational Geometry, 1991

Research paper thumbnail of Aproximación aleatoria de cuerpos convexos

Publicacions Matemàtiques, 1992

APROXIMACIÓN ALEATORIA DE CUERPOS CONVEXOS* FERNANDO AFFENTRANGER Problems related te the random ... more APROXIMACIÓN ALEATORIA DE CUERPOS CONVEXOS* FERNANDO AFFENTRANGER Problems related te the random approximation of convex bodies fall into the field of integral geometry and geometric probabilities. The aim of this paper is te give a survey of known results about the stochastic model that has received special attention in the literature and that can be described as follows : Let K be a d-dimensional convex body in Euclidean space Rd, d >_ 2. Denote by H the convex hull of n independent random points X 1,. .. , X distributed identically and uniformly in the interior of K. lf cp is a random variable en d-dimensional polytopes in Rd, we define the random variable cp, by ,pn = W(conv{X1,. .. ,Xn}), *Extended version of an invited talk,

Research paper thumbnail of Pairs of non-intersecting random flats

Probability Theory and Related Fields, 1988

Research paper thumbnail of Random circles in the d-dimensional unit ball

Journal of Applied Probability, 1989

This note gives the solution of the following problem concerning geometric probabilities. What is... more This note gives the solution of the following problem concerning geometric probabilities. What is the probability p(Bd; 2) that the circumference determined by three points P, P1 and P2 chosen independently and uniformly at random in the interior of a d-dimensional unit ball Bd in Euclidean space Ed (d ≧ 2) is entirely contained in Bd? From our result we conclude that p(Bd; 2) →π /(3√3) as d →∞.

Research paper thumbnail of Random projections of regular simplices

Discrete & Computational Geometry, 1992

Precise asymptotic formulae are obtained for the expected number of k-faces of the orthogonal pro... more Precise asymptotic formulae are obtained for the expected number of k-faces of the orthogonal projection of a regular n-simplex in n-space onto a randomly chosen isotropic subspace of fixed dimension or codimension, as the dimension n tends to infinity.

Research paper thumbnail of On the convex hull of uniform random points in a simpled-polytope

Discrete & Computational Geometry, 1991

Denote the expected number of facets and vertices and the expected volume of the convex hull Pn o... more Denote the expected number of facets and vertices and the expected volume of the convex hull Pn of n random points, selected independently and uniformly from the interior of a simple d-polytope by En(f), E.(v), and E~(V), respectively. In this note we determine the sharp constants of the asymptotic expansion of En(f), E.(v), and En(V), as n tends to infinity. Further, some results concerning the expected shape of P~ are given.

Research paper thumbnail of Random spheres in a convex body

Archiv der Mathematik, 1990

Research paper thumbnail of The Convex Hull of Random Balls

Advances in Applied Probability, 1993

While the convex hull of n d-dimensional balls is not a polytope, it does have an underlying comb... more While the convex hull of n d-dimensional balls is not a polytope, it does have an underlying combinatorial structure similar to that of a polytope. In the worst case, its combinatorial complexity can be of order Ω(n[d/2]). The thrust of this work is to show that its complexity is typically much smaller, and that it can therefore be constructed more quickly on average than in the worst case. To this end, four models of the random d-ball are developed, and the expected combinatorial complexity of the convex hull of n independent random d-balls is investigated for each. As n grows without bound, these expectations are O(1), O(n(d–1)/(d+4)), O(1) (for d = 2 only), and O(n(d–1)/(d+3)).

Research paper thumbnail of The expected volume of a random polytope in a ball*

Journal of Microscopy, 1988

For any convex body K in d‐dimensional Euclidean space Ed(d≥2) and for integers n and i, n ≥ d + ... more For any convex body K in d‐dimensional Euclidean space Ed(d≥2) and for integers n and i, n ≥ d + 1,1 ≤ i ≤ n, let V(d)n‐ii(K) be the expected volume of the convex hull Hn‐i, i of n independent random points, of which n‐i are uniformly distributed in the interior, the other i on the boundary of K. We develop an integral formula for V(d)n‐i, i(K) for the case that K is a d‐dimensional unit ball by considering an adequate decomposition of V(d)n‐i, i into d‐dimensional simplices. To solve the important case i = 0, that is the case in which all points are chosen at random from the interior of Bd, we require in addition Crofton's theorem on mean values. We illustrate the usefulness of our results by treating some special cases and by giving numerical values for the planar and the three‐dimensional cases.