Non-central sections of convex bodies (Q2408027)

Convex bodies are called admissibe if they have \(C^2\)-support functions. Let \(D\subset \mathrm{int} (K) \subset\mathbb R^n \) where \(D\) is a strictly convex and \(K\) is a convex body. Let \(H\) be a supporting plane to \(D\) at a point \(p=D\cap H \) with outer unit normal \(\xi\). Given \(u \in S^{n-1} \cap \xi^{\bot}\) we denote by \(\rho_{K,D} (u,\xi) = \rho_{K,p} (u)\) the radial function of \(K \cap H\) with respect to \(p\). Let \(H\) be a supporting plane to a convex body in \(\mathbb R^n\). We denote by \(H^+\) the half-space bounded by \(H\) and disjoint from the interior of \(D\). Given \(p \in \mathbb R^n\), \( v \in S^{n-1}\), let \(v_p^{\bot} = \{ x\in \mathbb R^n: \langle x-p, v \rangle = 0 \}\). \( v_p^+= \{ x\in\mathbb R^n: \langle x-p, v \rangle \geq 0 \}\). The main results of the paper are: Theorem 3.1. Let \(K\) and \(L\) be convex bodies in \(\mathbb R^2\) and let \(D_1\) and \(D_2\) be two admissible convex bodies in the interior of \(K\cap L\). If the chords \(K \cap H\) and \(L \cap H\) have equal lengths for all \(H\) supporting either \(D_1\) or \(D_2\), then \(K=L\). Theorem 3.2. Let \(K\), \(L\), \(D_1\), and \(D_2\) be as above. If \(\mathrm{vol}_2 (K \cap H^+) = \mathrm{vol}_2 (L\cap H^+)\) for every \(H\) supporting \(D_1\) or \(D_2\), then \(K=L\). Theorem 3.4. Let \(K\), \(L\), \(D_1\), and \(D_2\) be as above. Assume that for some \(i>0\) one of the following two conditions holds: (I) \( \rho^i_{K,D_j} (u, \xi) + \rho^i_{K,D_j} (- u, \xi) = \rho^i_{L,D_j} (u, \xi) + \rho^i_{L,D_j} (- u, \xi)\), for \(j=1,2\), (II) \(\partial K \cap \partial L \neq \emptyset\) and \(\rho^i_{K,D_j} (u, \xi) - \rho^i_{K,D_j} (- u, \xi) = \rho^i_{L,D_j} (u, \xi) - \rho^i_{L,D_j} (- u, \xi)\) for \(j=1,2\), for all \(\xi ,u \in S^1\) such that \(u \perp \xi\). Then \( K=L \). Theorem 4.1. Let \(K\) and \(L\) be convex bodies in \(\mathbb R^n \) (where \(n\) is even) and let \(D\) be a cube in the interior of \(K\cap L\). If \(\mathrm{vol}_{n-1} (K \cap H) = \mathrm{vol}_{n-1} (L\cap H)\) for any hyperplane passing through a vertex of \(D\) and an interior point of \(D\), then \(K=L\). The authors indicate that the case of odd dimensions in the Theorem 4.1. remains open, and that the smoothness assumptions for the support functions can be relaxed. Theorem 4.6. Let \(K\) and \(L\) be convex bodies in \(\mathbb R^n \), \(n\geq 3\), that contain a strictly convex body \(D\) in their interiors. Assume that \(\mathrm{vol}_{n-1} (K \cap H \cap v_p^+) = \mathrm{vol}_{n-1} (L\cap H \cap v_p^+)\) for every hyperplane \(H\) supporting \(D\) and every unit vector \(v \in H-p\), where \(p=D \cap H\). Then \(K=L\). Similar theorem holds for the half-sections \(K\cap H^+ \cap v^{\bot}\), \(L \cap H^+ \cap v^{\bot}\). Theorem 4.11. Let \(K\) and \(L\) be convex bodies in \(\mathbb R^n \), containing two distinct points \(p_1\) and \(p_2\) in their interiors. If, for every \(v \in S^{n-1}\), we have \(\mathrm{vol}_n (K \cap v_{p_j}^+) = \mathrm{vol}_n (L \cap v_{p_j}^+) \) for \(j=1,2\), then \(K=L\). Theorem 4.12. Let \(K\) and \(L\) be convex bodies of revolution in \(\mathbb R^n\) with the same axis of revolution. Let \(D\) be a convex body in the interior of both \(K\) and \(L\) such that \(D\) does not intersect the axis of revolution. If for any hyperplane \(H\) supporting \(D\) we have \(\mathrm{vol}_{n-1}(K\cap H)=\mathrm{vol}_{n-1}(L\cap H)\), then \(K=L\).