The floating body and the equiaffine inner parallel curve of a plane convex body (Q5933875)

Given a convex body \(K\) of area 1 in \(\mathbb{E}^2\), the author considers for \(0 < s < 1\) the convex floating body of \(K\) which is obtained by cutting off from \(K\) all segments of area \(s\) and the equiaffine inner parallel curve of \(K\) which is the envelope of all chords of \(K\) which cut off from \(K\) segments of area \(s\). Let \(K^s_{[s]}\) denote the difference of the area of \(K\) and the area of the convex floating body corresponding to \(s\) and let \(K_{[s]}\) be a certain sum of signed areas related to the equiaffine inner parallel curve corresponding to \(s\). Theorem 2: Let \(K\in {\mathcal C}^k_+\), \(k\geq 3\). Then there are constants \(a_2,\dots,a_{k-1}\), such that \(K_{[s]} = K^c_{[s]}=a_2s^{2/3} +\cdots + a_{k-1}s^{(k-1)/3} + o(s^{k/3})\) as \(s\to 0\). Theorem 3: Let \(K\in {\mathcal C}^k_+\), \(k\geq 3\). Then the expected area \(A_n(K)\) of the convex hull of \(n\) independently and uniformly distributed random points in \(K\) satisfies \(1 - A_n(K) = c_2n^{-2/3} +\cdots + c_kn^{-(k-1)/3} + o(n^{-k/3})\) as \(n\to\infty\), where \(c_2,\dots,c_{k-1}\) are suitable coefficients. The latter result extends a result of \textit{A. Rényi} and \textit{R. Sulanke} [Z. Wahrsch. Verw. Geb. 3, 138-147 (1964; Zbl 0126.34103)] and was first stated by \textit{P. Gruber} [Rend. Circ. Mat. Palermo (2), Suppl. 50, 189-216 (1997; Zbl 0896.52014)].