A convex cap, also known as a convex floating body[1] or just floating body,[2] is a well defined structure in mathematics commonly used in convex analysis for approximating convex shapes. In general it can be thought of as the intersection of a convex Polytope with a half-space.

Definition

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A cap,   can be defined as the intersection of a half-space   with a convex set  . Note that the cap can be defined in any dimensional space. Given a  ,   can be defined as the cap containing   corresponding to a half-space parallel to   with width   times greater than that of the original.

The definition of a cap can also be extended to define a cap of a point   where the cap   can be defined as the intersection of a convex set   with a half-space   containing  . The minimal cap of a point is a cap of   with  .[3][4]

Floating Bodies and Caps

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We can define the floating body of a convex shape   using the following process. Note the floating body is also convex. In the case of a 2-dimensional convex compact shape  , given some   where   is small. The floating body of this 2-dimensional shape is given by removing all the 2 dimensional caps of area   from the original body. The resulting shape will be our convex floating body  . We generalize this definition to n dimensions by starting with an n dimensional convex shape and removing caps in the corresponding dimension.

Relation to affine surface area

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As  , the floating body more closely approximates  . This information can tell us about the affine surface area   of   which measures how the boundary behaves in this situation. If we take the convex floating body of a shape, we notice that the distance from the boundary of the floating body to the boundary of the convex shape is related to the convex shape's curvature. Specifically, convex shapes with higher curvature have a higher distance between the two boundaries. Taking a look at the difference in the areas of the original body and the floating body as  . Using the relation between curvature and distance, we can deduce that   is also dependent on the curvature. Thus,

 .[5]

In this formula,   is the curvature of   at   and   is the length of the curve.

We can generalize distance, area and volume for n dimensions using the Hausdorff measure. This definition, then works for all  . As well, the power of   is related to the inverse of   where   is the number of dimensions. So, the affine surface area for an n-dimensional convex shape is

 

where   is the  -dimensional Hausdorff measure.[5]

Wet part of a convex body

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The wet part of a convex body can be defined as   where   is any real number describing the maximum volume of the wet part and  .[3][4]

We can see that using a non-degenerate linear transformation (one whose matrix is invertible) preserves any properties of  . So, we can say that   is equivariant under these types of transformations. Using this notation,  . Note that

 

is also equivariant under non-degenerate linear transformations.

Caps for approximation

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Assume   and choose   randomly, independently and according to the uniform distribution from  . Then,   is a random polytope.[3] Intuitively, it is clear that as  ,   approaches  . We can determine how well   approximates   in various measures of approximation, but we mainly focus on the volume. So, we define  , when   refers to the expected value. We use   as the wet part of   and   as the floating body of  . The following theorem states that the general principle governing   is of the same order as the magnitude of the volume of the wet part with  .

Theorem

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For   and  ,  .[3] The proof of this theorem is based on the technique of M-regions and cap coverings. We can use the minimal cap which is a cap   containing   and satisfying  . Although the minimal cap is not unique, this doesn't have an effect on the proof of the theorem.

Lemma

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If   and  , then   for every minimal cap  .[3]

Since  , this lemma establishes the equivalence of the M-regions   and a minimal cap  : a blown up copy of   contains   and a blown up copy of   contains  . Thus, M-regions and minimal caps can be interchanged freely, without losing more than a constant factor in estimates.

Economic cap covering

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A cap covering can be defined as the set of caps that completely cover some boundary  . By minimizing the size of each cap, we can minimize the size of the set of caps and create a new set. This set of caps with minimal volume is called an economic cap covering and can be explicitly defined as the set of caps   covering some boundary   where each   has some minimal width   and the total volume of this covering is ≪   .[3]

References

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  1. ^ Besau, Florian; Werner, Elisabeth M. (October 2016). "The spherical convex floating body". Advances in Mathematics. 301: 867–901. arXiv:1411.7664. doi:10.1016/j.aim.2016.07.001. ISSN 0001-8708.
  2. ^ M., Nagy, Stanislav Schütt, Carsten Werner, Elisabeth (2019). Halfspace depth and floating body. The American Statistical Association, the Bernoulli Society, the Institute of Mathematical Statistics, and the Statistical Society of Canada. OCLC 1108755798.{{cite book}}: CS1 maint: multiple names: authors list (link)
  3. ^ a b c d e f Bárány, Imre (2007). "Random Polytopes, Convex Bodies, and Approximation" (PDF). Stochastic Geometry: Lectures Given at the C.I.M.E. Summer School Held in Martina Franca, Italy, September 13–18, 2004. Lecture Notes in Mathematics. Vol. 1892. Springer. pp. 77–118. doi:10.1007/978-3-540-38175-4_2. ISBN 978-3-540-38174-7.
  4. ^ a b "Floating Bodies - Numberphile". YouTube.
  5. ^ a b Ludwig, Monika; Reitzner, Matthias (15 October 1999). "A Characterization of Affine Surface Area". Advances in Mathematics. 147 (1): 138–172. doi:10.1006/aima.1999.1832. ISSN 0001-8708.