x_1, x_2, x_3, ..., x_n are n positive real numbers whose sum S is constant
the sum of their squares shuld be as smallas possible
what are the n numbers?

x_1, x_2, x_3, ..., x_n are n positive real numbers whose sum S is constant
the sum of their squares shuld be as smallas possible
what are the n numbers?
We can write the n numbers as
where .
The sum of their squares is
Hence the sum of squares is minimized when , i.e. when .
Thus the n numbers are .
That is nice tight reasoning. I like it.Originally Posted by JaneBennet
Here is a way to think about the problem geometrically. The problem is scalable, so take S =1. Mininizing the sum of the squares is equivalent to minimizing the square root of the sum of the squares.
Now, first think about the case n=2. The locus of the points whose absolute value sums to 1 is a square with vertices at (1.0), (0.1), (1,0), and (1.1). We are only interested in what happens in the upper right hand quadrant, but think about the whole square. Now the square root of the sum of the squares is the ordinary Eudlidean distance to the origin. Geometrically you can see that this is minimized for points on the square at the points that also lie on lines with slope 1 or 1, i.e. where , If you do this same exercise in higher dimensions you find that the locus of points whose absolute value sums to a constant is just a hypercube and the solution is still in the center point of each hypersurface, which is where the absolute values of the invividual coordinates are equal.
Your solution is tighter and more rigorously argued, but I think the intuition from the geometric version is helpful in figuring out what the answer ought to be.
Well, I solved a problem similar to this very recently, so I was able to use the same method as before.
But it’s always nice to be able to generalize a solution, to understand why a problem is a particular case of a more general situation.
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