On convex intersection bodies and unique determination problems for convex bodies (Q298123)

Let \(L \subset {\mathbb R}^n\) be a star body. Its intersection body is the star body \(IL \subset {\mathbb R}^n\) with the radial function NEWLINE\[NEWLINE\rho_{IL} (\xi) = \text{vol}_{n-1} (L \cap \xi^{\bot}), \qquad \xi \in S^{n-1}.NEWLINE\]NEWLINE The cross-section body of a convex body \(K \subset {\mathbb R}^n\) is the star body \(CK \subset {\mathbb R}^n\) with NEWLINE\[NEWLINE\rho_{CK} = \max_{t \in {\mathbb R}} \, \text{vol}_{n-1} (K \cap (\xi^{\bot} + t \xi)), \qquad \xi \in S^{n-1}.NEWLINE\]NEWLINE Let \(g = g(K) \in K\) be the centroid of \(K\), and \(K^{*y}\) be the polar body of \(K\) with respect to \(y \in \text{int} K\): NEWLINE\[NEWLINEK^{*y} = \{ x \in {\mathbb R}^n \mid \langle x-y, z-y \rangle \leq 1 \quad \forall z \in K \}.NEWLINE\]NEWLINE The convex intersection body of \(K\) is the (a priori) star body \(CI(K) \subset {\mathbb R}^n\) with NEWLINE\[NEWLINE\rho_{CI(K)} (\xi) = \min \left\{ \text{vol }_{n-1} \left[(K^{*g} \mid \xi^{\bot})^{*y} \right] \mid y \in \text{int} (K^{*g} \mid \xi^{\bot}) \right\} , \quad \xi \in S^{n-1}NEWLINE\]NEWLINE where \(\cdot \mid \xi^{\bot}\) is the orthogonal projection onto the hyperplane perpendicular to \(\xi\).NEWLINENEWLINELet \(K \mapsto \widetilde{K}\) be a map from the set of convex bodies to the set of star bodies in \({\mathbb R}^n\) such that: {\parindent=6mm \begin{itemize}\item[{\(\bullet\)}] \(\widetilde{K}\) is always origin-symmetric; \item[{\(\bullet\)}] \(IK = \widetilde{K}\) for all origin-symmetric convex \(K \subset {\mathbb R}^n\); \item[{\(\bullet\)}] There is a sequence \(\{K_m \}\) of convex non-centrally-symmetric bodies in \({\mathbb R}^n\) such that \(\{ \widetilde{K}_m \}\) are infinitely smooth and NEWLINE\[NEWLINE\lim_{m \to \infty} \| \rho_{\widetilde{K}_m } - a \|_{C^k (S^{n-1})} = 0 \quad \forall \, \, \text{natural} \, k,NEWLINE\]NEWLINE where \(a > 0\) is a constant independent of \(k\). NEWLINENEWLINE\end{itemize}} NEWLINEThe main results of the paper are connected with counter-examples constructions:NEWLINENEWLINETheorem 2. Let \(n \geq 2\). There are infinitely smooth convex bodies of rotation \(K, L \subset {\mathbb R}^n\) such that \(K\) is not centrally-symmetric, \(L\) is origin-symmetric, and \(CI (K) = CI (L)\).NEWLINENEWLINETheorem 5. There exists a non-centrally-symmetric convex body \(K\) and an (infinitely smooth) origin-symmetric convex body \(L\) such that \(\widetilde{K} = \widetilde{L}\). Namely, take \(L = L_m\) defined by NEWLINE\[NEWLINE\rho_{L_m} = \left[(n-1) {\mathcal R}^{-1} \rho_{\widetilde{K}_m} \right]^{\frac{1}{n-1}},NEWLINE\]NEWLINE and \(K = K_m\) for large enough \(m\).NEWLINENEWLINEHere \({\mathcal R}\) is the spherical Radon transform. The author formulates a naturalNEWLINENEWLINEQuestion. Let \(n \geq 3\). Is there a convex body \(K\subset {\mathbb R}^n\) which is not centrally-symmetric, and whose convex intersection body \(CI(K)\) is a Euclidean ball?