This inequality holds on Hilbert Spaces? I am trying to solve a probelm and if I conclude that the following inequality holds I finish it.
Consider $H$ an Hilbert space over $\mathbb{R}$. Let $\alpha_1, ..., \alpha_n$ be real numbers such that $\alpha_i\geq 0$, for all $i=1,...,n$ and $\sum_{i=1}^n\alpha_i=1$. Let $x_1, ...,x_n \in H$ and define
$$x:=\sum_{i=1}^n\alpha_ix_i$$
My question is: the following inequality holds for all $i,j$?
$$\lVert x-x_i\rVert ^2+\lVert x-x_j\rVert ^2 \leq \lVert x_i-x_j\rVert ^2$$
I can easily see that it holds when $\{x_1, ..., x_n\}$ is orthogonal, but I'm not sure in the general case. I would appreciate any hint. Thanks!
 A: Geometric Answer: The vectors $x_i$ span an at most $n$ dimensional subspace of $H$, so we can ignore the rest of the (possibly infinite dimensional) Hilbert space and just work in $n$-dimensional Euclidean space. If the inequality in the question were false for all $i\neq j$, that would mean, by the law of cosines, that all the angles $\angle x_ixx_j$ are acute. But then, if we fix any paricular index $i$, we would have all the $x_j$'s (including $x_i$) strictly on the same side of the hyperplane through $x$ perpendicular to $x-x_i$. That's absurd, since $x$, which is on that hyperplane, is a weighted average (with weights $\alpha_j$) of the $x_j$'s.
Algebraic Answer: To simplify notation, let $y_i=x_i-x$. So $\sum_i\alpha_iy_i=0$. The left side of  your inequality is $\Vert y_i\Vert^2+\Vert y_j\Vert^2$, and the right side is, when you expand the norm in terms of the inner product, $\Vert y_i\Vert^2+\Vert y_j\Vert^2-2(y_i\cdot y_j)$.  So if your inequality fails, you'd have $(y_i\cdot y_j)>0$. Fix some $i$, sum over $j$, and remember that $\sum_j\alpha_jy_j=0$, to get the contradiction $0>0$.
Note that both proofs show that, for each $i$, there is at least one $j$ such that your inequality is satisfied.
Note also that, in the geometric proof, the simplification of working in finite-dimensional spaces isn't really needed provided you know the law of cosines and the connection between half-spaces and weighted averages in infinite-dimensional Hilbert spaces.
