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Inner approximations of convex sets and intersections of projectionally exposed cones

Bruno F. Lourenço, Vera Roshchina, James Saunderson

TL;DR

The paper addresses whether the intersection of projectionally exposed cones preserves $p$-exposedness and shows that this is false in dimension $5$ by constructing two $\mathbb{R}^5$ cones $\mathcal{K}_1,\mathcal{K}_2$ that are $p$-exposed but have $\mathcal{K}_1\cap\mathcal{K}_2$ not $p$-exposed; it also provides the first amenable cone that is not $p$-exposed. A core contribution is the development of an inner-approximation toolkit (face-fixing inner approximations and sandwiching) that can tightly approximate a compact convex set while fixing a given face and simplifying the remainder of the facial structure. This framework enables turning local counterexamples into global ones, by constructing modified cones $\tilde{\mathcal{K}}_i$ whose faces are either extreme rays or the fixed faces, preserving $p$-exposure for those faces while ensuring the intersection loses $p$-exposure. Together, these results separate projectional exposure from amenability and niceness in dimension five and provide a versatile method potentially applicable to spectrahedral and hyperbolicity cones.

Abstract

A convex cone is said to be projectionally exposed (p-exposed) if every face arises as a projection of the original cone. It is known that, in dimension at most four, the intersection of two p-exposed cones is again p-exposed. In this paper we construct two p-exposed cones in dimension $5$ whose intersection is not p-exposed. This construction also leads to the first example of an amenable cone that is not projectionally exposed, showing that these properties, which coincide in dimension at most $4$, are distinct in dimension $5$. In order to achieve these goals, we develop a new technique for constructing arbitrarily tight inner convex approximations of compact convex sets with desired facial structure. These inner approximations have the property that all proper faces are extreme points, with the exception of a specific exposed face of the original set.

Inner approximations of convex sets and intersections of projectionally exposed cones

TL;DR

The paper addresses whether the intersection of projectionally exposed cones preserves -exposedness and shows that this is false in dimension by constructing two cones that are -exposed but have not -exposed; it also provides the first amenable cone that is not -exposed. A core contribution is the development of an inner-approximation toolkit (face-fixing inner approximations and sandwiching) that can tightly approximate a compact convex set while fixing a given face and simplifying the remainder of the facial structure. This framework enables turning local counterexamples into global ones, by constructing modified cones whose faces are either extreme rays or the fixed faces, preserving -exposure for those faces while ensuring the intersection loses -exposure. Together, these results separate projectional exposure from amenability and niceness in dimension five and provide a versatile method potentially applicable to spectrahedral and hyperbolicity cones.

Abstract

A convex cone is said to be projectionally exposed (p-exposed) if every face arises as a projection of the original cone. It is known that, in dimension at most four, the intersection of two p-exposed cones is again p-exposed. In this paper we construct two p-exposed cones in dimension whose intersection is not p-exposed. This construction also leads to the first example of an amenable cone that is not projectionally exposed, showing that these properties, which coincide in dimension at most , are distinct in dimension . In order to achieve these goals, we develop a new technique for constructing arbitrarily tight inner convex approximations of compact convex sets with desired facial structure. These inner approximations have the property that all proper faces are extreme points, with the exception of a specific exposed face of the original set.
Paper Structure (10 sections, 14 theorems, 93 equations, 2 figures)

This paper contains 10 sections, 14 theorems, 93 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Notions of exposedness
  4. Approximating convex sets
  5. Face-fixing inner approximations of convex sets
  6. Sandwiching convex sets
  7. Intersections of projectionally exposed cones
  8. The key construction
  9. Proof of Theorem \ref{['thm:main']}
  10. Conclusions and Some Open Questions

Key Result

Theorem 1.1

For any compact convex set $C\subset\mathop{\mathrm{\mathbb{R}}}\nolimits^n$, a proper nonempty exposed face $F$ of $C$ and any continuous function $\varphi: C\to \mathop{\mathrm{\mathbb{R}}}\nolimits_+$ such that $\varphi(x)=0$ if and only if $x\in F$, there exists a compact convex set ${D}\subsete Here by $\partial C$ we denote the relative boundary of $C$.

Figures (2)

  • Figure 1: Left: the projections of the curves $\alpha,\beta$ and $\gamma$ onto the slice of the face $F_1$ of ${\mathcal{K}}_1$; right: the $xyz$-projections of these curves, along with the projection of the slice of the face $F_1$ of the cone ${\mathcal{K}}_1$
  • Figure 2: A diagram showing the construction used in proof of Theorem \ref{['thm:main']}

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 18 more