Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 45, No. 1, pp. 155167 (2004) 

Deviation measures and normals of convex bodiesH. GroemerDepartment of Mathematics, The University of Arizona, Tucson, AZ 85721, USA, email: groemer@email.arizona.eduAbstract: With any given convex body we associate three numbers that exhibit, respectively, its deviation from a ball, a centrally symmetric body, and a body of constant width. Several properties of these deviation measures are studied. Then, noting that these special bodies may be defined in terms of their normals, corresponding deviation measures for normals are introduced. Several inequalities are proved that show that convex bodies cannot deviate much from these special types if their corresponding deviations of the normals are small. These inequalities can be interpreted as stability results. Full text of the article:
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