# Some comments on floating and centroid bodies in the plane

Some comments on floating and centroid bodies in the plane Consider a long, convex, homogenous cylinder with horizontal axis and with a planar convex body K as transversal section. Suppose the cylinder is immersed in water and let \$\$K_w\$\$ K w be the wet part of K. In this paper we study some properties of the locus of the centroid of \$\$K_w\$\$ K w and prove an analogous result to Klamkin–Flanders’ theorem when the locus is a circle. We also study properties of bodies floating at equilibrium when either the origin or the centroid of the body is pinned at the water line. In some sense this is the floating body problem for a density varying continuously. Finally, in the last section we give an isoperimetric type inequality for the perimeter of the centroid body (defined by C. M. Petty in Pacific J Math 11:1535–1547, 1961) of convex bodies in the plane. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png aequationes mathematicae Springer Journals

# Some comments on floating and centroid bodies in the plane

, Volume 92 (2) – Jan 12, 2018
12 pages

Publisher
Springer International Publishing
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Analysis; Combinatorics
ISSN
0001-9054
eISSN
1420-8903
D.O.I.
10.1007/s00010-017-0525-4
Publisher site
See Article on Publisher Site

### Abstract

Consider a long, convex, homogenous cylinder with horizontal axis and with a planar convex body K as transversal section. Suppose the cylinder is immersed in water and let \$\$K_w\$\$ K w be the wet part of K. In this paper we study some properties of the locus of the centroid of \$\$K_w\$\$ K w and prove an analogous result to Klamkin–Flanders’ theorem when the locus is a circle. We also study properties of bodies floating at equilibrium when either the origin or the centroid of the body is pinned at the water line. In some sense this is the floating body problem for a density varying continuously. Finally, in the last section we give an isoperimetric type inequality for the perimeter of the centroid body (defined by C. M. Petty in Pacific J Math 11:1535–1547, 1961) of convex bodies in the plane.

### Journal

aequationes mathematicaeSpringer Journals

Published: Jan 12, 2018

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