Aequat. Math. 92 (2018), 211–222
Springer International Publishing AG,
part of Springer Nature 2018
published online January 12, 2018
Some comments on ﬂoating and centroid bodies in the plane
Z. Guerrero-Zarazua and J. Jer
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
be the wet part of K. In this paper we study some properties of the locus of
the centroid of K
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.
Mathematics Subject Classiﬁcation. 52A10, 53A04.
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
be the wet part and dry part of K, respectively.
According to Archimedes’ law, K ﬂoats in equilibrium if the line joining the
center of gravity, g
, of the wet part and the center of gravity g of K,is
orthogonal to the water line. Since we know that the center of gravity g
the dry part is collinear with g and g
, we may say that K ﬂoats in equilibrium
if the segment [g
] is orthogonal to the water line xy, as shown in Fig. 1.
Moreover, if a body K ﬂoats in equilibrium at an arbitrary orientation and
the volume of the wet part is a fraction δ of the volume of K we say that
K ﬂoats in equilibrium at every position with density δ. There is a famous
problem due to S. Ulam in the Scottish book , which asks: is a solid of
uniform density which ﬂoats in water in every position a Euclidean ball? The
2-dimensional version of this problem is: if a long, convex, homogenous cylinder
ﬂoats in equilibrium at every position (with horizontal axis), is its cross-section
necessarily a Euclidean disc? The answer to this 2-dimensional problem is