Say $K$ is a $n$dimensional convex subset of $\mathbb R^n$ around the origin. Say we know $Vol_{n  1}(\pi_{\theta^{\bot}}) K)$ where $\theta^\bot$ is the linear subspace of $\mathbb{R}^n$ orthogonal to the point $\theta$ of $S^{n  1}$, and $\pi_{\theta^{\bot}}$ is orthogonal projection onto that subspace; also, $Vol_{n  1}$ is Lebesgue measure on $\mathbb{R}^{n  1}$. Can we write $Vol_{n}(K)$ as an integral over $Vol_{n  1}(\pi_{\theta^{\bot}}) K)$ (and say its derivatives with respect to $\theta$ etc). It is ok to include some kind of diameter function in the formula of $Vol_{n}(K)$.

If I understand the question correctly, no. In particular, for a curve of constant width, the width does not determine the area. 


Have a look at:
[JFM 34.0649.01 Minkowski, H.
Volumen und Oberfläche. (German)
Math. Ann. 57, 447495 (1903)] and
[MR0478079 (57 #17572) 

