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.

这是一篇关于星形基本数学概念的说明性论文，着重强调了星形集的几何，分析，组合和拓扑性质及其在许多数学领域中的广泛应用。作者决定以非常广泛的方式来研究该主题，因为他们不知道有任何相关的类似调查的出版物涉及这种自然概念。星形的概念与凸的概念非常接近，在经典凸，凸分析，泛函分析，离散，组合和计算几何，微分几何，逼近理论，PDE和优化等领域中都需要使用。它与径向函数，截面函数，可见性，（支撑）圆锥，内核，对偶性等概念紧密相关。我们以详细的方式介绍了星型集合的基本性质的许多定义和定理，然后是相关结果的类似调查的讨论。在本文的结尾，我们还对上述某些学科中的广泛应用进行了调查。