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Báo cáo hóa học: "COMPARING THE RELATIVE VOLUME WITH THE RELATIVE INRADIUS AND THE RELATIVE WIDTH"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: COMPARING THE RELATIVE VOLUME WITH THE RELATIVE INRADIUS AND THE RELATIVE WIDTH | COMPARING THE RELATIVE VOLUME WITH THE RELATIVE INRADIUS AND THE RELATIVE WIDTH A. CERDAN Received 28 February 2006 Revised 24 August 2006 Accepted 29 August 2006 We consider subdivisions of a convex body G in two subsets E and G E. We obtain several inequalities comparing the relative volume 1 with the minimum relative inradius 2 with the maximum relative inradius 3 with the minimum relative width and 4 with the maximum relative width. In each case we obtain the best upper and lower estimates for subdivisions determined by general hypersurfaces and by hyperplanes. Copyright 2006 A. Cerdan. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Let G be an open bounded convex set. Let us consider subdivisions of G in two connected subsets E and G E in such a way that the relative boundary dE n G is a connected topological hypersurface. Relative geometric inequalities compare functionals that give information about the geometry of the subdivision we call these functionals relative geometric measures. This kind of inequalities provides either lower or upper estimates of the ratio between appropriate powers of two of those relative geometric measures. In case that these upper or lower estimates exist we call the sets for which the equality sign is attained maximizers or minimizers . The oldest relative geometric inequalities that were investigated are the so-called relative isoperimetric inequalities. They provide upper estimates of the ratio between appropriate powers of the relative volume the minimum between the volume of E and the volume of its complement and the relative perimeter the length of the relative boundary . Many results about relative isoperimetric inequalities are included in 3 5 . Recently some results have been obtained by comparing the relative perimeter and the minimum relative