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In this paper, we find a way to give best simultaneous approximation of n arbitrary points in convex sets. First, we introduce a special hyperplane which is based on those n points. Then by using this hyperplane, we define best approximation of each point and achieve our purpose.

As we known, best approximation theory has many applications. One of the best results is best simultaneous approximation of a bounded set but this target cannot be achieved easily. Frank Deutsch in [

In [3,4] we can see that a hyperplane of an n-dimensional space is a flat subset with dimension.

In this paper we try to find best simultaneous approximation of n arbitrary points in convex sets. We say theorems of best approximation of a point in convex sets.

Then we give the method of finding best simultaneous approximation of n points in convex set.

In this paper, we consider that X is a real inner product space. For a nonempty subset W of X and , define

.

Recall that a point is a best approximation of if

If each has at least one best approximation , then W is called proximinal.

We denote by , the set of all best approximations of x in W. Therefore

It is well-known that is a closed and bounded subset of X. If, then is located in the boundary of W.

In 2.4 lemma of [

In particular, every closed convex subset K of a Hilbert space X has a unique best approximation in K.

Also in 4.1 lemma of [

For a nonempty subset W of X and a nonempty bounded set S in X, define

and

Each element in (If) is called a best simultaneous approximation to S from W (see [

For and hyperplane H in X defined by

and we denote H by.

The Kernel of a functional f is the set

and for

we say that is in the below of hyperplane H, if .

In this section,we consider

and

Define

Lemma 3.1. Let consider the hyperplane then

Proof. Give so we have

So by adding with equation of above, we have

Therefore have

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Note 3.2. It is obvious that . Now let , so there exist such that for all

Thus , therefore w will be in W_{i}, that we conclude

Theorem 3.3. Let then:

1)

2) If W be a convex subset of X, then W_{i} is a convex set.

3) If W be a closed set, then W_{i} is a closed set.

Proof. 1) Let therefore

so then we have

so by adding with equation of above, we have

therefore we have

.

Thus we have

.

Therefore.

Since all previous steps will be reversible, so for any in a fixed i, we have that consider

thus we have

so

therefore

and finally

.

2) First we proof , for all is convex set.

Give and , set

thus we have

So . Thus is convex set and since intersection of any convex set is convex, therefore W_{i} is convex set.

3) It is obviously that f is continuous function and we know

.

So, is closed set, this implies W_{i} is closed set. ■

The following theorem states that to find best simultaneous approximation of a bounded set S of W, it is enough to obtain the best approximation to any

.

Thus would be the best simultaneous approximation of S from W if is minimal.

Theorem 4.1. If W be a convex subset of X and there exist for all , then

Proof. With attention of best simultaneous approximation and (3.2) notation, we have

but according to the definition of W_{i} we have

thus the above equation can be written as follows

and since exist

so we have

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Corollary 4.2. With the assumptions of the previous theorem there exist i, such that is best simultaneous approximation of S in W.

Proof. With attention previous theorem, there exist such that

and by the definition of we have

after according to the above equation and define the best simultaneous approximation of the relationship will

However, the algorithm with assumes a convex set W and introduce the following.

With attention 3.1 lemma for points x_{1}, x_{2} the hyperplane are possible to obtain, by 3.4 definition the points W in below H_{12}_{} are V_{12}_{} called.

Also for points x_{1}, x_{3} the hyperplane

are formed and the points of W in below H_{13}_{} are V_{13} called and so we do order to the points x_{1}, x_{n}.

By taking subscribe of any, find W_{1} that this set is convex (by Theorem 3.3, 2).

Therefore, if best approximation x_{1} exists in this set, it is called . Thus obtain for any

.

Finally, the point which has minimal distance to x_{i}, is the best simultaneous approximation of S in W.