Let $v_1,...,v_n\in \mathbb{R}^n$ be linearly independent. The parallelepiped defined by these vectors is $P(v_1,...,v_n)=\{\sum_{i=1}^{n}\alpha_i v_i|~0\le\alpha_i\le 1\}$. Observe that while the collection $v_1,...,v_n$ is reconstructible from $P(v_1,...,v_n)$ as a subset of $\mathbb{R}^n$, it is not reconstructible from $P(v_1,...,v_n)$ as a geometric figure. Indeed, if $U$ is an orthogonal operator on $\mathbb{R}^n$, then $P(Uv_1,...,Uv_n)$ and $P(v_1,...,v_n)$ are isometric. Moreover, $P(v_1,...,-v_i,...,v_n)$ is a translate of $P(v_1,...,v_n)$ by $-2v_i$. You can also view this as the shift of origin: instead of looking from the point $0$ we are now looking from the point $v_i$. In fact $P(\pm v_1,...,\pm v_n)$ correspond to every vertex of this parallelepiped.

For $A=\{i_1,...,i_k\}\subset\{1,...,n\}$ define $V(A)=V_k(P(v_{i_1},...,v_{i_k}))$, where $V_k$ is the $k$-dimensional volume.

Let $w_1,...,w_n\in \mathbb{R}^n$ also be linearly independent, and for $A\subset\{1,...,n\}$ define $W(A)$ analogously.

I can show the following

Proposition. If $W(A)=V(A)$ for every $A\subset\{1,...,n\}$, then $P(w_1,...,w_n)$ and $P(v_1,...,v_n)$ are isometric, i.e. there are $a_1,...,a_n=\pm 1$ and an orthogonal operator $U$ on $\mathbb{R}^n$, such that $w_i=a_iUv_i$, for every $i$.

However, I can only do it using a result about principal minors of a symmetric matrix determining it up to multiplying both $i$-th row and $i$-th column by $\pm 1$ (in our case we consider the Gram matrices, whose principal minors are exactly the squares of the corresponding volumes).

Is the Proposition known? Is there a geometric proof of it?

I tried to build a proof on the fact that we know the distance from every $v_i$ to the span of any other combination of $v_j$ (including the empty one), but geometry kind of gets intertwined with combinatoric of what is orthogonal to what, and I got stuck.

PS In fact the Proposition is equivalent the result that I've mentioned (in the real case), and it is proven e.g. here:

Rising, Justin; Kulesza, Alex; Taskar, Ben, An efficient algorithm for the symmetric principal minor assignment problem, Linear Algebra Appl. 473, 126-144 (2015). ZBL1314.65050.

Let $v_1,...,v_n\in \mathbb{R}^n$ be linearly independent. The parallelepiped defined by these vectors is $P(v_1,...,v_n)=\{\sum_{i=1}^{n}\alpha_i v_i|~0\le\alpha_i\le 1\}$. Observe that while the collection $v_1,...,v_n$ is reconstructible from $P(v_1,...,v_n)$ as a subset of $\mathbb{R}^n$, it is not reconstructible from $P(v_1,...,v_n)$ as a geometric figure. Indeed, if $U$ is an orthogonal operator on $\mathbb{R}^n$, then $P(Uv_1,...,Uv_n)=UP(v_1,...,v_n)$ is isometric to $P(v_1,...,v_n)$. Moreover, $P(v_1,...,-v_i,...,v_n)$ is a translate of $P(v_1,...,v_n)$ by $-v_i$. You can also view this as the shift of origin: instead of looking from the point $0$ we are now looking from the vertex $v_i$. In fact $P(\pm v_1,...,\pm v_n)$ correspond to every of $2^{n}$ vertex of this parallelepiped.

For $A=\{i_1,...,i_k\}\subset\{1,...,n\}$ define $V(A)=V_k(P(v_{i_1},...,v_{i_k}))$, where $V_k$ is the $k$-dimensional volume.

Let $w_1,...,w_n\in \mathbb{R}^n$ also be linearly independent, and for $A\subset\{1,...,n\}$ define $W(A)$ analogously.

I can show the following

Proposition. If $W(A)=V(A)$ for every $A\subset\{1,...,n\}$, then $P(w_1,...,w_n)$ and $P(v_1,...,v_n)$ are isometric, i.e. there are $a_1,...,a_n=\pm 1$ and an orthogonal operator $U$ on $\mathbb{R}^n,$ such that $w_i=a_iUv_i$, for every $i$.

However, I can only do it using a result about principal minors of a symmetric matrix determining it up to multiplying both $i$-th row and $i$-th column by $\pm 1$ (in our case we consider the Gram matrices, whose principal minors are exactly the squares of the corresponding volumes).

Is the Proposition known? Is there a geometric proof of it?

I tried to build a proof on the fact that we know the distance from every $v_i$ to the span of any other combination of $v_j$ (including the empty one), but geometry kind of gets intertwined with combinatoric of what is orthogonal to what, and I got stuck.

PS In fact the Proposition is equivalent the result that I've mentioned (in the real case), and it is proven e.g. here:

Rising, Justin; Kulesza, Alex; Taskar, Ben, An efficient algorithm for the symmetric principal minor assignment problem, Linear Algebra Appl. 473, 126-144 (2015). ZBL1314.65050.