On the Monotonicity of the Isoperimetric Quotient for Parallel Bodies

GND
120652196
Affiliation
Institute of Mathematics, Friedrich Schiller University
Richter, Christian;
ORCID
0000-0002-1986-9641
Affiliation
ALTA Institute for Algebra, Geometry, Topology and their Applications, University of Bremen, Bremen, Germany
Saorín Gómez, Eugenia

The isoperimetric quotient of the whole family of inner and outer parallel bodies of a convex body is shown to be decreasing in the parameter of definition of parallel bodies, along with a characterization of those convex bodies for which that quotient happens to be constant on some interval within its domain. This is obtained relative to arbitrary gauge bodies, having the classical Euclidean setting as a particular case. Similar results are established for different families of Wulff shapes that are closely related to parallel bodies. These give rise to solutions of isoperimetric-type problems. Furthermore, new results on the monotonicity of quotients of other quermassintegrals different from surface area and volume, for the family of parallel bodies, are obtained.

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