Covariograms of convex bodies in the plane: A remark on Nagel's theorem (Q699846)

Let \(K\) be a compact convex subset of the Euclidean plane and let \(\lambda\) be the Lebesgue measure in \({\mathbb E}^2\). The family of areas \(\{\lambda (K\cap (K+v))\}_{v\in {\mathbb R}^2}\) is called the covariogram of \(K\). It has been shown by \textit{W. Nagel} [J. Appl. Probab. 30, 730-736 (1993; Zbl 0781.60018)] that if \(K\) is a convex polygon then \(K\) is uniquely determined by its covariogram up to translation and reflection with respect to a point. \textit{G. Bianchi, F. Segala}, and \textit{A. Volcic} [Quad. Matematici Dip. Sci. Mat. Universita di Trieste 485 (2000)] have generalized Nagel's theorem to the case where \(K\) is a compact convex body with a piecewise \(C_+^2\)-boundary. This statement does not hold in higher dimensions; there are known counterexamples in dimensions \(d\geq 4\). In this paper the author proves the following theorem. Let \(K_1\) and \(K_2\) be point-symmetric compact convex bodies in the plane whose boundaries are piecewise two times continuously differentiable. Then, if the supports of the covariograms \(c(K_1)\) and \(c(K_2)\) have the same boundary \(D\) and if \(c(K_1)\) and \(c(K_2)\) coincide in a neighborhood of \(D\), it follows that \(K_2=K_1+w\) for some \(w\in {\mathbb R}^2\).