Starshaped sets

Abstract

This is an expository paper about the fundamental mathematical notion of starshapedness, emphasizing the geometric, analytical, combinatorial, and topological properties of starshaped sets and their broad applicability in many mathematical fields. The authors decided to approach the topic in a very broad way since they are not aware of any related survey-like publications dealing with this natural notion. The concept of starshapedness is very close to that of convexity, and it is needed in fields like classical convexity, convex analysis, functional analysis, discrete, combinatorial and computational geometry, differential geometry, approximation theory, PDE, and optimization; it is strongly related to notions like radial functions, section functions, visibility, (support) cones, kernels, duality, and many others. We present in a detailed way many definitions of and theorems on the basic properties of starshaped sets, followed by survey-like discussions of related results. At the end of the article, we additionally survey a broad spectrum of applications in some of the above mentioned disciplines.

Intersection bodies of polytopes

“…Moreover, Ludwig showed that the unique non-trivial GL(d)-covariant star-body-valued valuation on convex polytopes corresponds to taking the intersection body of the dual polytope (Ludwig 2006). Due to such results, the knowledge on properties of intersection bodies interestingly contributes also to the (still not systematized) theory of starshaped sets, see Section 17 of the exposition (Hansen et al 2020). Recently, there is increased interest in investigating convex geometry from an algebraic point of view (Blekherman et al 2013;Sinn 2015;Rostalski and Sturmfels 2010;Ranestad and Sturmfels 2011).…”

Sensitivity to large losses and $ρ$-arbitrage for convex risk measures

“…The goal of this appendix is to state some results that relate star-shaped sets/functions (about the origin) with their corresponding recession cone/function. For a recent survey on star-shaped sets, see [33]. The definition of star-shaped functions varies in the literature.…”

Equilibrium points, periodic solutions and the Brouwer fixed point theorem for convex and non-convex domains

“…In particular, a continuous map φ such that φ(∂A) ⊂ A has a fixed point [15, p. 33] (see also [62,Corollary 1]). For other fixed point theorems in the setting of star-shaped sets, see [28,Section 19.5] and [49]. Starshaped sets with the stronger condition at the boundary considered in Definition 3.1 are also studied in different contexts.…”

“…The assumption that the set G is strictly star-shaped not only with respect to a point p, but also with respect to all the points in a neighborhood of p, is an hypothesis which is rather common in the theory of star-shaped sets and is usually referred saying that the strong kernel of G has nonempty interior (cf. [28]). Actually, if the open set G has a kernel with nonempty interior (according to [28, p. 1005]), that is G is star-shaped with respect to all the points of (small) open ball B ⊂ G then, according to [28, Theorem 3, p. 1006], for each p ∈ B, and u ∈ ∂G, it follows that [p, u[ ⊂ G. As a consequence, ∂G = ∂G and G is strictly star-shaped with respect to p. Therefore, G is strictly star-shaped also with respect to all the points in a neighborhood of p. Using the positions x 1 := t, x 2 := x, we see that the positive semi-orbits of system (3.12) are the graphs of the solutions (t, x(t)) of ẋ = ϕ(x) satisfying the initial condition x(t 0 ) = x 0 , for t ≥ t 0 in the (t, x)-plane.…”