Not all convex functions on $\mathbb{R}^n$ have finite minimizers; some can only be minimized by a sequence as it heads to infinity. In this work, we aim to develop a theory for understanding such minimizers at infinity. We study astral space, a compact extension of $\mathbb{R}^n$ to which such points at infinity have been added. Astral space is constructed to be as small as possible while still ensuring that all linear functions can be continuously extended to the new space. Although astral space includes all of $\mathbb{R}^n$, it is not a vector space, nor even a metric space. However, it is sufficiently well-structured to allow useful and meaningful extensions of concepts of convexity, conjugacy, and subdifferentials. We develop these concepts and analyze various properties of convex functions on astral space, including the detailed structure of their minimizers, exact characterizations of continuity, and convergence of descent algorithms.

翻译：并非所有定义在 $\mathbb{R}^n$ 上的凸函数都具有有限极小值点；某些函数只能通过趋于无穷远的序列实现极小化。本研究旨在建立一套理论以理解此类位于无穷远处的极小值点。我们研究星体空间——一种将 $\mathbb{R}^n$ 紧致化并添加了无穷远点的扩展空间。星体空间的构造追求最小化扩展规模，同时确保所有线性函数都能连续延拓至该新空间。尽管星体空间包含整个 $\mathbb{R}^n$，但它既非向量空间，亦非度量空间。然而，其结构具有足够的良好性质，使得凸性、共轭性与次微分等概念能够获得有用且意义明确的延拓。我们系统发展了这些概念，并分析了星体空间上凸函数的各种性质，包括其极小值点的精细结构、连续性的精确刻画以及下降算法的收敛性。