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Extended Set Difference : Inverse Operation of Minkowski Summation

Arie Beresteanu, Behrooz Moosavi Ramezanzadeh

TL;DR

The paper addresses the lack of a genuine inverse for Minkowski summation by introducing the extended set difference $A \ominus_e B$, defined as the collection of minimizers of the Hausdorff distance $d_H(A, B \oplus X)$ over $X \in \mathcal{K}^d_{kc}$. Existence of minimizers is established, and the authors analyze properties, symmetry, convergence, and bounds, while presenting a unique solution strategy via strictly convex perturbations. A practical, LP-based computation framework in the space of support functions is developed, enabling efficient approximation of $A \ominus_e B$ by convex polygons and successful application to multiple geometric configurations. The approach provides a robust, constructive tool for set-valued arithmetic in higher dimensions with potential applications in optimization, computer vision, and geometric analysis.

Abstract

This paper introduces the extended set difference, a generalization of the Hukuhara and generalized Hukuhara differences, defined for compact convex sets in $\mathbb{R}^d$. The proposed difference guarantees existence for any pair of such sets, offering a broader framework for set arithmetic. The difference may not be necessarily unique, but we offer a bound on the variety of solutions. The definition of the extended set difference is formulated through an optimization problem, which provides a constructive approach to its computation. The paper explores the properties of this new difference, including its stability under orthogonal transformations and its robustness to perturbations of the input sets. We propose a method to compute this difference through a formulated linear optimization problem.

Extended Set Difference : Inverse Operation of Minkowski Summation

TL;DR

The paper addresses the lack of a genuine inverse for Minkowski summation by introducing the extended set difference , defined as the collection of minimizers of the Hausdorff distance over . Existence of minimizers is established, and the authors analyze properties, symmetry, convergence, and bounds, while presenting a unique solution strategy via strictly convex perturbations. A practical, LP-based computation framework in the space of support functions is developed, enabling efficient approximation of by convex polygons and successful application to multiple geometric configurations. The approach provides a robust, constructive tool for set-valued arithmetic in higher dimensions with potential applications in optimization, computer vision, and geometric analysis.

Abstract

This paper introduces the extended set difference, a generalization of the Hukuhara and generalized Hukuhara differences, defined for compact convex sets in . The proposed difference guarantees existence for any pair of such sets, offering a broader framework for set arithmetic. The difference may not be necessarily unique, but we offer a bound on the variety of solutions. The definition of the extended set difference is formulated through an optimization problem, which provides a constructive approach to its computation. The paper explores the properties of this new difference, including its stability under orthogonal transformations and its robustness to perturbations of the input sets. We propose a method to compute this difference through a formulated linear optimization problem.
Paper Structure (17 sections, 15 theorems, 92 equations, 2 figures, 1 algorithm)

This paper contains 17 sections, 15 theorems, 92 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notations
  3. The structure of the paper
  4. Extended Set Difference
  5. Existence
  6. Properties of Set Differences
  7. Basic Properties
  8. Orthonormal Transformations
  9. Convergence
  10. Scope
  11. Symmetry
  12. Uniqueness Refinement
  13. Computation of the Extended Set Difference
  14. LP Formulation
  15. Computational Method
  16. ...and 2 more sections

Key Result

Theorem 1

Let $A, B \in \mathcal{K}^d_{kc}$. Then there exists a set $X \in \mathcal{K}^d_{kc}$ such that $X \in A \ominus_e B.$

Figures (2)

  • Figure 1: Example \ref{['ex:ballsquare']}: (a) a circle $A$, (b) a square $B$, and (c) the derived set $X^{\ast} \in A \ominus_e B$.
  • Figure 2: Example \ref{['ex:polygons']}: (a) a pentagon $A$, (b) a square $B$, and (c) the derived set $X^{\ast} \in A \ominus_e B$.

Theorems & Definitions (24)

  • Definition 1: Extended Set Difference
  • Theorem 1: Existence
  • Proposition 2: Basic Properties
  • Proposition 3
  • Proposition 4
  • Proposition 5: Convergence
  • Theorem 6: Scope
  • Theorem 7
  • Example 1
  • Definition 2
  • ...and 14 more