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Reduced convex bodies in Euclidean space – a survey. (English) Zbl 1213.52006

Summary: A convex body \(R\) in Euclidean space \(E^d\) is called reduced if the minimal width \(\Delta (K)\) of each convex body \(K \subset R\) different from \(R\) is smaller than \(\Delta (R)\). This definition yields a class of convex bodies which contains the class of complete sets, i.e., the family of bodies of constant width. Other obvious examples in \(E^{2}\) are regular odd-gons. We know a relatively large amount on reduced convex bodies in \(E^{2}\). Besides theorems which permit us to understand the shape of their boundaries, we have estimates of the diameter, perimeter and area. For \(d\geq 3\) we do not even have tools which permit us to recognize what the boundary of \(R\) looks like. The class of reduced convex bodies has interesting applications.
We present the current state of knowledge about reduced convex bodies in \(E^d\), recall some striking related research problems, and put a few new questions.

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A21 Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry)
52A38 Length, area, volume and convex sets (aspects of convex geometry)
52A40 Inequalities and extremum problems involving convexity in convex geometry
52B12 Special polytopes (linear programming, centrally symmetric, etc.)
52B60 Isoperimetric problems for polytopes
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