Im R^r seien n Punkte a1 …. . an gegeben. Zeige, dal3 die Summe der Abstandsquadrate f(x)=∑v=1^n

step 1 Hello everyone, let us assume that x of i is equal to m i minus a where a is some number. Now p

Im R^r seien n Punkte a1 …. . an gegeben. Zeige, dal3 die...

Im $\mathrm{R}^{r}$ seien $n$ Punkte $a_{1 \ldots . .} a_{n}$ gegeben. Zeige, dal3 die Summe der Abstandsquadrate $$ f(x)=\sum_{v=1}^{n}\left|x-a_{2}\right|^{2} \quad\left(x \in \mathcal{R}^{p}\right) $$ ein Minimum im, Mittelpunkt" $\xi:=\frac{1}{n} \sum_{n-1}^{n} a_{i}$ des Punktsystems $a_{1} \ldots \ldots a_{n}$ besitzt. $V_{g l}$ auch

Im $\mathrm{R}^{r}$ seien $n$ Punkte $a_{1 \ldots . .} a_{n}$ gegeben. Zeige, dal3 die Summe der Abstandsquadrate
$$
f(x)=\sum_{v=1}^{n}\left|x-a_{2}\right|^{2} \quad\left(x \in \mathcal{R}^{p}\right)
$$
ein Minimum im, Mittelpunkt" $\xi:=\frac{1}{n} \sum_{n-1}^{n} a_{i}$ des Punktsystems $a_{1} \ldots \ldots a_{n}$ besitzt. $V_{g l}$ auch

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Key Concepts

Quadratic Optimization

The problem involves minimizing a quadratic function, which is expressed as the sum of squared Euclidean distances from a variable point to a collection of fixed points. Quadratic functions are well-known in optimization due to their convexity properties, ensuring a unique, global minimum.

Centroid

The centroid of a set of points, often called the arithmetic mean, is the point obtained by averaging all the individual coordinates of the points. In many optimization problems, including least-squares minimization, the centroid uniquely minimizes the sum of squared distances to the given points.

Convexity

A function is convex if any local minimum is also a global minimum, which is a key property when optimizing quadratic functions. The strict convexity of the sum of squared distances ensures that the identified minimum, which occurs at the centroid, is indeed the unique global minimum.

Differentiation in Euclidean Space

The process of finding the minimum of a differentiable function involves computing its gradient and setting it to zero. This calculus-based method confirms that the minimum of the sum of squared distances occurs when the variable point equals the centroid, as the derivative leads directly to this conclusion.

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