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Suppose I have $n$ vectors in $\mathbb R^n, \{x_1,\ldots,x_n\}$ s.t. any $x_k, 1\leq k\leq n$ "opens" a new dimension. (For example, no $4$ vectors of $(x_i)$ lie on a same plane, and bo 3 vectors lie on the same line, etc).

Is there a name in such a case to $\text{conv}\{x_1\ldots,x_n\}$ for some general $n$?

For example, if $n=1$ then this is a dot. If $n=2$ this is a segment. If $n=3$ this is a triangle.

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You are describing the simplex in each dimension.

https://en.wikipedia.org/wiki/Simplex

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