We consider the distance function (DF), given by the caliber (the Minkowski gauge function) of a convex body, from a point to strictly, strongly, and weakly convex sets in an arbitrary Hilbert space. Some properties of the caliber of a strongly convex set and the conditions for obtaining a strict, strong, or weak convexity of Lebesgue sets for the distance function are established in accordance with the requirements for the set, the caliber of which specifies the distance function, and the set to which the distance is measured. The corresponding inequalities are obtained that reflect the behavior of the distance function on segments and allow comparing it with strictly, strongly, or weakly convex functions.

中文翻译：

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非对称空间中强弱凸集距离函数的性质

我们考虑了距离函数（DF），它是由凸体的口径（Minkowski规范函数）给出的，它是从任意点到任意希尔伯特空间中的严格，强和弱凸集的距离。根据集的要求，建立了强凸集的口径的某些性质以及为距离函数获得Lebesgue集的严格，强或弱凸度的条件，该条件的口径指定了距离函数，以及要测量距离的集合。获得了相应的不等式，这些不等式反映了距离函数在段上的行为，并允许将其与严格，强或弱凸函数进行比较。