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In this paper, we prove the relationship between selection expectation and support function by a new method.

The studies of random geometrical objects can go back at least to the famous Buffon needle problem [

The relationship between random sets and convex geometry has been thoroughly explored within the stochastic geometry literature; see, e.g. Weil and Wieacker [

The organization of this manuscript is as follows. In the next section, we set notations and give preliminaries. In the last section we will prove the relationship between selection expectation and support function by a new method.

We consider the d-dimensional Euclidean space

Let

For

Obviously, for K,

Hence a convex body is uniquely determined by its support function.

In order to show that a sequence of the sets of nonempty, compact subsets converges to another set of nonempty, compact subsets. One must define the distance between two sets. It motivates the following definition.

For

Then

The point

The set of all convex combinations of any finitely many elements of A is called the convex hull of A and is denoted by coA.

The family of closed subsets of

A map

It is natural to define random open sets as complements to random closed sets, so that

Therefore we can regard a random set X as a measurable map defined on an abstract probability space

A random set X is called simple, if there exists a finite measurable partition

A random set X is called approximable if X is an almost sure limit of a sequence of simple random sets.

Similarly, a random set X with almost surely compact values is called a random compact set.

The norm

A random vector

The space

Following Artstein and Vitale [

Remark: 1) Selection expectation (also called the Aumann expectation), which is the best investigated concept of expectation for random sets. Since many results can be naturally formulated for random closed sets in Banach spaces.

2) We cite an example to illustrate the meaning of selection expectation. If X is a simple random compact set, i.e.

3)

4) Moreover, if the probability space

Now we are ready to prove the main result in this section.

Theorem ([

Proof. One of two probability spaces contains the atoms which may lead to the fact that two independent and identically distributed random compact sets may have different selection expectations. Therefore if

Note that if X is a random compact set, then

For

Let

It follows that

The authors would like to acknowledge the support from the Educational Commission of Hunan Province of China (12A033) and the Hunan Provinial Natural Science Foundation of China (14JJ2122).

RigaoHe, (2015) A Note on the Selection Expectation and Support Function. Advances in Pure Mathematics,05,583-586. doi: 10.4236/apm.2015.510055