The authors of this paper begin with a description of the well-known Busemann-Petty problem (see [H. Busemann and C. M. Petty, Math. Scand. 4, 88–94 (1956; Zbl 0070.39301)]) and give an interesting review of the literature existing in this field till date by discussing a number of approaches adopted by various authors to study the various facets of this problem. In a recent paper, G. Giannopoulos and A. Koldobsky [Trans. Am. Math. Soc. 370, No. 6, 4351–4372 (2018; Zbl 1387.52014)] proved the inequality \[ V_d{(K)^{\frac{k}{d}}} - V_d{(L)^{\frac{k}{d}}} \le {r^{d - k}} \max_{H \in G(d,k)} (V_k(K \cap H) - V_k(L \cap H)) \] for \(K,L\) origin-symmetric convex bodies in \(\mathbb{R}^d\) such that \(L\subset K\) and \(G(d,k)\) is the Grassmanian of \(k\)-dimensional subspace of \(\mathbb{R}^d\) by “estimating the distance between volumes of two convex bodies in terms of difference between areas of their sections”. In the present paper, the authors, “instead of comparing the volumes between convex bodies,” investigate the stability and determination of convex bodies by taking into account the Hausdorff distance between convex bodies and by posing the problem as below:
“If \(K\) and \(L\) are convex bodies containing the origin in \(\mathbb{R}^d\), and for each \((d-1)\)-dimensional subspace \(H\) and some \(0<\varepsilon<1\) satisfy \(\left| {V_{d-1}(K \cap H) - V_{d-1}(L \cap H)} \right|< \varepsilon\), does there exist a constant \(C\) such that the Hausdorff distance between \(K\) and \(L\) satisfies \(\delta (K,L) < C\varepsilon\)?”
Without considering the symmetry assumption of the star bodies, the authors establish “that a star body \(K\) is uniquely determined by the volumes of sections and its half volumes \(V(K \cap u^+)\), for \(u \in S^{d-1}\) and \(u^+ = \left\{{x:x \in \mathbb{R}^d,x \cdot u \ge 0} \right\}\)” and the following core result is proven in this connection:
Theorem 1. (Main) Let \(K\) and \(L\) be two star bodies with respect to the origin in \(\mathbb{R}^d\). If they have the same volumes of their central sections and half volumes, then \(K = L\).
The following stability version of the above result is also proven by the authors for convex bodies:
Theorem 2. (Main) Let \(K,L \in \mathscr{K}^d(r,R), d \ge 3\). If, for any \(u \in S^{d-1}\) and some \(0 \le \varepsilon \le 1\), \(\left\| {V(K \cap u^+) - V(L \cap u^+)} \right\| \le \varepsilon\) and \(\left\| {V(K \cap u^\bot) - V(L \cap u^\bot)} \right\| \le \varepsilon\), then \(\delta (K,L) \le c(d,r,R){\varepsilon^{\frac{4}{{(d + 1)(d + 4)}}}}\) with an explicit constant \(c(d, r,R)\) depending only on \(d, r,R\).
It is also pertinent to point out here that in an earlier work of H. Groemer [Monatsh. Math. 126, No. 2, 117–124 (1998; Zbl 0918.52002)]) it was shown that a \(d\)-dimensional star body \(K\) with respect to the origin \(o\) can be uniquely determined from the volume of intersections by half-planes that contain \(o\) on the boundary with the body \(K\) and the uniqueness and stability results were proved there by considering the hemispherical transformation \(\mathscr{T}_{d-2}\) on the \((d-2)\)-dimensional unit sphere \(S(u)=S^{d-1}\cap u^\bot\), whereas in the present paper the authors have employed the spherical Radon transformation \(\mathscr{R}\) and the hemispherical transformation \(\mathscr{T}\) on \((d-1)\)-dimensional unit sphere \(S^{d-1}\) to prove the above two results. Thus, the above results are not deducible from the results of Groemer [loc. cit.]. Still another interesting feature of this study is that the stability result of Theorem 2 above is proven in this paper by estimating the distance between convex bodies by Hausdorff distance whereas Giannopoulos and Koldobsky [loc. cit.] deduced their inequalities by “estimating the distance between volumes of two convex bodies”.