Suppose that we have a list of vectors, where each vector's norm is 3.
Then, if we average these vectors (per element), the final vector's norm will be $\leq$ 3 .
This is only empirically speaking. I also noticed that more vectors i average whose norm is 3 for example, the closer I get to 3 as an upper bound. Is there some sort of proof for this or was this random?
Using the triangle identity and absolute homogeneity (which are included in the definition of a norm), if $\forall k\in\{ 0,\dots,n\}, \mathcal{N}(u_k)=v$: $$ \mathcal{N}(\frac{1}{n}\sum_{k=0}^nu_k)\leq \frac{1}{n}\sum_{k=0}^n\mathcal{N}(u_k) = v$$
As to the 'getting closer' part, this is not true in general: take the norm $||v||_1 = \sum_{i=0}^1 |v_i|$ in $\mathbb{R}^2$, $v_1 = (1,0)^T, v_2=-v_1$.