Suppose that $n$ people $A_1$, $A_2$, $\ldots$, $A_n$, ($n \geq 3$) are seated in a circle and that $A_i$ has $a_i$ objects such that 
\[ a_1 + a_2 + \cdots + a_n = nN \] 
where $N$ is a positive integer. In order that each person has the same number of objects, each person $A_i$ is to give or to receive a certain number of objects to or from its two neighbours $A_{i-1}$ and $A_{i+1}$. (Here $A_{n+1}$ means $A_1$ and $A_n$ means $A_0$.)  How should this redistribution be performed so that the total number of objects transferred is minimum?
1. **Define the problem in terms of variables:**
   Let \( a_i \) be the number of objects person \( A_i \) has, and we are given that:
   \[
   a_1 + a_2 + \cdots + a_n = nN
   \]
   where \( N \) is a positive integer. We need to redistribute the objects so that each person ends up with \( N \) objects.

2. **Express the redistribution in terms of transfers:**
   Let \( x_i \) be the number of objects person \( A_i \) gives to person \( A_{i+1} \). Note that \( x_i \) can be negative, indicating that \( A_i \) receives objects from \( A_{i+1} \).

3. **Set up the equation for redistribution:**
   After redistribution, each person \( A_i \) should have \( N \) objects. Therefore, the number of objects person \( A_i \) has after redistribution can be expressed as:
   \[
   a_i + x_{i-1} - x_i = N
   \]
   Rearranging this equation, we get:
   \[
   x_i = x_{i-1} + a_i - N
   \]

4. **Determine the total cost of redistribution:**
   The total cost of redistribution is the sum of the absolute values of the transfers:
   \[
   \sum_{i=1}^n |x_i|
   \]

5. **Express \( x_i \) in terms of \( x_1 \):**
   From the equation \( x_i = x_{i-1} + a_i - N \), we can express each \( x_i \) in terms of \( x_1 \):
   \[
   x_2 = x_1 + a_1 - N
   \]
   \[
   x_3 = x_2 + a_2 - N = x_1 + (a_1 - N) + (a_2 - N)
   \]
   Continuing this pattern, we get:
   \[
   x_i = x_1 + \sum_{j=1}^{i-1} (a_j - N)
   \]

6. **Minimize the total cost:**
   To minimize the total cost \( \sum_{i=1}^n |x_i| \), we need to find the optimal value of \( x_1 \). The cost function \( \sum_{i=1}^n |x_1 - b_i| \) is minimized when \( x_1 \) is the median of the values \( b_i \), where:
   \[
   b_i = \sum_{j=1}^{i-1} (a_j - N)
   \]

7. **Conclusion:**
   The optimal redistribution is achieved by setting \( x_1 \) to the median of the values \( b_i \). This ensures that the total number of objects transferred is minimized.

The final answer is \( \boxed{ x_1 } \) should be the median of the values \( \sum_{j=1}^{i-1} (a_j - N) \).