Showing papers on "Minkowski addition published in 1988"
01 Jan 1988
TL;DR: The product of the volumes of a symmetric zonoid A in RI and of its polar body is minimal if and only if A is the Minkowski sum of n segments as discussed by the authors.
Abstract: A new and simple proof of the following result is given: The product of the volumes of a symmetric zonoid A in RI and of its polar body is minimal if and only if A is the Minkowski sum of n segments.
94 citations
01 Dec 1988-Graphical Models \/graphical Models and Image Processing \/computer Vision, Graphics, and Image Processing
TL;DR: Using two new shape operators called Minkowski addition and decomposition operators, a simple shape model is presented and mathematical characteristics of these operators are explored in some detail, with the aim of eventually arriving at a formal theory of shape description.
Abstract: Using two new shape operators called Minkowski addition and decomposition operators, a simple shape model is presented. Mathematical characteristics of these operators are explored in some detail, with the aim of eventually arriving at a formal theory of shape description. A few application areas of the shape model, particularly some important uses of the shape operators are briefly mentioned.
69 citations
TL;DR: In this article, it was shown that if we approximate the Euclidean ball by a Minkowski sum of n segments, then the smallest possible n is equal (up to a possible logarithmic factor) toc(n)e −2(n−1)/(n+2).
Abstract: It is proved that if we approximate the Euclidean ballB
n in the Hausdorff distance up toɛ by a Minkowski sum ofN segments, then the smallest possibleN is equal (up to a possible logarithmic factor) toc(n)e
−2(n−1)/(n+2). A similar result is proved ifB
n is replaced by a general zonoid inR
n
.
54 citations
TL;DR: For a finite set of points in a Minkowski space, it is shown in this paper that a solution can be constructed by solving a linear programming problem, at least approximately.
Abstract: For a finite set of points in a Minkowski space a point has to be found such that the sum of the distances between this point and the points of the set is as small as possible. It is shown that a solution can be constructed by solving a linear programming problem, at least approximately.
19 citations
TL;DR: In this paper, the surface area of a convex body bounded by an ellipsoid with principal axes of lengths 2a, 2b, and 2c was derived in terms of the surface areas of balls or spheroids.
Abstract: Given the convex body E=E(a,b,c) bounded by the ellipsoid with principal axes of lengths 2a, 2b, and 2c, its surface area, S(a,b,c), is a non-elementary integral unless a=b=c, (E is a ball) or two values of a,b, and c are equal (E is a solid spheroid). This leads to upper and lower estimates for S(a,b,c) in terms of the surface areas of balls or spheroids. We derive many of the known inequalities and some new inequalities for the surface areas of ellipsoids using Minkowski sums of ellipsoids and Minkowski's mixed volumes.
2 citations