J. Geom. 87 (2007) 99 – 105 0047–2468/07/020099 – 7 © Birkh¨ auser Verlag, Basel, 2007 DOI 10.1007/s00022-006-1905-4 Minimal area n-simplex circumscribing a strictly convex body in R n Andrzej Miernowski, Witold Mozgawa and Witold Rzymowski Abstract. In this paper we are interested in characterization of tangency points of a minimal area n-simplex circumscribed about a strictly convex compact body in R n . Mathematics Subject Classification (2000): 52A20, 53A07. Key words: simplex, Lagrange multiplier, circumscribing, homothety 1. Introduction Mark Levi in [3] considered minimal perimeter triangles circumscribing a strictly convex set in the plane with a regular boundary. The author proved that for the minimal perimeter triangle the segments connecting the vertices of the triangle with the opposite tangency points are concurrent, as well as the lines perpendicular to the sides of the triangle at these points. The similar theorem is well-known for the minimal area triangle circumscribing a strictly convex set in the plane ([1], [3]). Note, that in this case the tangency points are the midpoints of the sides of the triangle ([6], [2]). The similar statement holds also for minimal volume tetrahedron circumscribing a strictly convex body in R 3 ([2]). On the other hand the formulas of Levi (see also [6]) show that tangency points of a minimal perimeter triangle are the tangency points of its excircles. The segments connecting the vertices with these points concur at the Nagel point of the triangle. In the paper [4] we characterized the tangency points of a minimal area tetrahedron circum- scribed about a strictly convex compact body in R 3 . In this paper we are interested if there is a similar geometric characterization of tangency points of a minimal area n-simplex circumscribed about a strictly convex compact body in R n . We prove the following theorem: THEOREM 1.1. Let S(x 0 ,x 1 ,...,x n ) be a minimal area n-simplex with vertices x 0 ,x 1 ,..., x n circumscribed about a strictly convex compact body C ⊂ R n with the nonempty inte- rior. Let J 1 1-n c x 0 ,J 1 1-n c x 1 ,...,J 1 1-n c xn be the homotheties with ratio 1 1-n and centers c x 0 , c x 1 , ..., c x n at centroids of the corresponding faces  (x 1 ,x 2 ,...,x n ),  (x 0 ,x 2 ,...,x n ),...,  (x 0 ,x 1 ,...,x n-1 ). Then the tangency points of C with the simplex S(x 0 ,x 1 ,...,x n ) 99