Determination of convex bodies by certain sets of sectional volumes (Q5951928)

Investigating a variant of Hammer's X-ray problem [\textit{P. C. Hammer}, Proc. Symp. Pure Math. VII: Convexity, Providence, RI, AMS, 498-499 (1963), Problem 2] the authors conjecture: Suppose \(K_1, K_2\), and \(L\) are convex bodies in \(\mathbb{R}^d\) with \(L\subset \text{int } K_1\cap \text{int }K_2\). If for all hyperplanes \(H\) that support \(L\) the \((d -1)\)-volumes \(|K_1\cap H|=|K_2 \cap H|\), must then \(K_1 = K_2\)? For the case when \(L=B\), the unit ball, a number of partial results are presented. NEWLINENEWLINENEWLINEIn two dimensions, one is a characterization of circle: If the unit circle \(B\subset K\), and all tangents of \(B\) intersect \(K\) in segments ot equal length, then \(K\) is a circle. Another considers the case when the unit circle \(B\) touches the boundary \(\partial K\) of \(K\). Suppose that \(B\subset K_1\), \(K_2\subset\mathbb{R}^2\), and \(\partial K_1\cap B\) and \(\partial K_2\cap B\) are single points. If the lengths of the chords \(K_1\cap l\) and \(K_2\cap l\) are equal for all lines \(l\) supporting \(B\), then \(K_1 = K_2\). And a condition is derived from the lemma: If \(B\subset\text{int }K\), and \(P\) and \(Q\) are polygons inscribed in \(K\) with edges supporting \(B\), then \(P\) and \(Q\) have the same number of vertices: \(N(K)\). From this the authors derive a necessary condition for the conjecture not to hold: If \(K_1\neq K_2\), with \(B\subset\text{int }K_1\cap\text{int} K_2\), and the lengths \(|K_1\cap l|= |K_2\cap l|\) whenever \(l\) is a line supporting \(B\), then \(N(K_1)\) and \(N(K_2)\) must be equal and even.NEWLINENEWLINENEWLINEIn higher dimensions, let \(K_1\), \(K_2\) be convex bodies and \(B\subset \text{int } K_1\cap \text{int } K_2\). Suppose that for some \(j\), \(1\leq j \leq d-2\), every \(j\)-flat \(H\) that supports \(B\) intersects \(K_1\) and \(K_2\) in sets of equal \(j\)-volume; then \(K_1 = K_2\). As in two dimensions, no complete answer is known for \(j = d-1\). The \((d-1)\)-volumes of the sections \(H\cap K\) determine \(K\), however, if \(H\) ranges over the hyperplanes supporting an infinity of concentric balls (that the proof given works only for odd \(d\) is probably not significant).