The convex floating body of a convex body 
is the intersection
of the closed half-spaces left after cutting off caps
of a prescribed volume from 
(Schütt and Werner 1990, p. 276).
Write 
for 
-dimensional
Lebesgue measure, and suppose 
and 
. For each unit
vector 
,
let 
be the unique real number
such that 
,
where 
is the Euclidean inner product and 
is the Euclidean l2-norm.
Then the convex floating body is
 |
Here 
is an absolute cap volume, rather than a fraction of 
. If a relative parameter 
is used instead, the corresponding absolute parameter
is 
.
The intersection is a compact convex set contained in 
, but it may be the empty set.
In particular, it need not itself be a convex body
with nonempty interior.
For a triangle of area 
, it is empty when 
. Indeed, cuts parallel
to the three sides require each of the normalized barycentric
coordinates to be at most 
, whereas their sum
is one. Increasing 
shrinks the retained half-spaces, so 
when 
.
If 
is an invertible affine transformation,
then 
,
since every cap volume is multiplied by 
, where 
is the determinant of 
. Thus the relative-volume parameter
is unchanged by invertible affine transformations.
For a planar disk of radius 
, rotational
symmetry makes 
the concentric disk of radius 
, where 
is determined by the circular
segment area
 |
Here 
denotes the inverse cosine. The equation
has a unique solution for 
, because the cap area
decreases strictly from 
to zero as 
increases from zero to 
.
The intersection construction is distinct from requiring every supporting hyperplane of the resulting
set to cut off volume 
from 
. The latter is the classical floating-body condition and does
not follow merely from the intersection definition
(Schütt and Werner 1990, pp. 276-277).
