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<p>The standard closed convex hull of a set is defined as the intersection of all images, under the action of a group of rigid motions, of a half-space containing the given set. In this paper we propose a generalisation of this classical notion, that we call a (<i>K</i>, ℍ)-hull, and which is obtained from the above construction by replacing a half-space with some other closed convex subset <i>K</i> of the Euclidean space, and a group of rigid motions by a subset ℍ of the group of invertible affine transformations. The main focus is on the analysis of (<i>K</i>, ℍ)-convex hulls of random samples from <i>K</i>.</p>

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Generalised convexity with respect to families of affine maps

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  • 出版模式：

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  • 发表日期：

    2025年4月3日

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  • Zakhar Kabluchko,
  • Alexander Marynych,
  • Ilya Molchanov

摘要

The standard closed convex hull of a set is defined as the intersection of all images, under the action of a group of rigid motions, of a half-space containing the given set. In this paper we propose a generalisation of this classical notion, that we call a (K, ℍ)-hull, and which is obtained from the above construction by replacing a half-space with some other closed convex subset K of the Euclidean space, and a group of rigid motions by a subset ℍ of the group of invertible affine transformations. The main focus is on the analysis of (K, ℍ)-convex hulls of random samples from K.