We are given a $d$-dimensional convex shape $S$ inscribed in the hypercube $[-1,1]^d$. We want find an approximation of its volume based only on a set of curves given by the intersection of the $S$ boundary and a finite number of $2$-planes.


We denote by $\gamma\_{i,j}$ the curve given by the intersection of the $S$ boundary with the $2$-plane $\mathrm{span}\left(\mathbf{e}\_i,\mathbf{e}\_j\right)$ for all $1\le i<j\le d$, where $\mathbf{e}\_1,\mathbf{e}\_2,\ldots,\mathbf{e}\_d$ are the standard basis vectors.




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**Question:** Knowing only the curves $\gamma\_{i,j}$ for all $1\le i<j\le d$, is it possible to provide an estimation $V'$ of the volume $V$ of $S$ such that there exist two positive constants $\alpha$ and $\beta$ for which $\alpha\le\frac{V'}{V}\le\beta$?



Estimating the volume of a convex shape in higher dimensions based only on normal sections
Those constants don't exist for any $d\geq4$, here is an idea of why.


For each $\varepsilon>0$ let $A\_\varepsilon=\{(x\_1,\dots,x\_d)\in[-1,1]^d;\lvert (d-1)x\_d-\sum\_{i=1}^{d-1} x\_i\rvert\leq\varepsilon\text{ and }\lvert\frac{(d-1)(d-2)}{2}x\_d-\sum\_{i=1}^{d-1}ix\_i\rvert\leq\varepsilon\}$. Note that there is a big constant $K$ independent of $\varepsilon$ such that for all $i,j$, the intersection of $A\_\varepsilon$ with span$(e\_i,e\_j)$ is contained in the ball $B(0,K\varepsilon)$.


Also note that $v=(1,1,\dots,1)\in A\_\varepsilon$, and let $B\_\varepsilon=\{x\in A\_\varepsilon;d(x,span(v))\leq K\varepsilon\}$.


Then both $A\_\varepsilon$ and $B\_\varepsilon$ are convex and inscribed in the cube, and their intersection with span$(e\_i,e\_j)$ is the same $\forall i,j$, but when $\varepsilon\to0$, the volume of $A\_\varepsilon$ is, up to some constant, proportional to $\varepsilon^2$ and the volume of $B\_\varepsilon$ is proportional up to some constant to $\varepsilon^{d-1}$.


