It is a classical result that a star-shaped body in \(\mathbb R^d\) symmetric with respect to the origin \(o\) is uniquely determined by the \((d-1)\)-volumes of its hyperplane sections through \(o\). This is not true anymore if the star body is not centrally symmetric.
In this paper a quick proof is given that a compact body starshaped with respect to \(o\) is uniquely determined by the \((d-1)\)-volumes and the centroids of its hyperplane sections through \(o\). The main part of the paper is devoted to proving an \(L_2\) stability version of this result for convex bodies. In short, if the volume and centroid functions of two convex bodies are close, then the radial functions of the two convex bodies are also close, where closeness is defined in terms of the \(L_2\) norm. More precisely, let \(\mathcal K^d(r,R)\) denote the collection of all convex bodies \(K\) in \(\mathbb R^d\) such that \(rB^d\subseteq K\subseteq RB^d\), where \(B^d\) is the Euclidean unit ball. Let \(\rho_K\) denote the radial function of \(K\) with respect to \(o\). For a unit vector \(u\), let \(v_{d-1}(K,u)\) and \(c_{d-1}(K,u)\) denote the \((d-1)\)-volume and centroid, respectively, of the section of \(K\) with the hyperplane through \(o\) orthogonal to \(u\). Let \(\varepsilon_0>0\). Then there exists a \(c=c(d,r,R,\varepsilon_0)\) such that for any \(\varepsilon\in[0,\varepsilon_0)\) and any \(K,L\in\mathcal K^d(r,R)\), if \(\| v_{d-1}(K,\cdot)-v_{d-1}(L,\cdot)\| _2\leq\varepsilon\) and \(\| c_{d-1}(K,\cdot)-c_{d-1}(L,\cdot)\| _2\leq\varepsilon\), then \(\| \rho_K-\rho_L\| \leq c\varepsilon^{2/d}\), where the \(L_2\) norms are defined by integration over the unit sphere with respect to spherical Lebesgue measure. The proof uses a stability result for the spherical Radon transform on even functions.