Planar convex bodies equidistant to all symmetric convex bodiesMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

The n-dimensional simplex is a convex body well-known for its numerous remarkable features and it was studied extensively also from the point of view of the Banach-Mazur distance. Our starting point is the following well-known and interesting property: the n-dimensional simplex is equidistant (in the Banach-Mazur distance) to all symmetric convex bodies with the distance equal to n. Moreover, it is known the simplex is the unique convex body with this property. It is therefore natural to ask if the simplex is the unique convex body that is equidistant to all symmetric convex bodies, but not necessarily with the distance equal to n. We answer this question negatively in the planar case. For all 7/4 < r < 2 we provide a general construction of a family of convex bodies K, which are with the distance r to every symmetric convex body. It should be noted that this distance r coincides with the asymmetry constant of K and the construction is based on some basic properties of the asymmetry constant.