On the homothety conjecture (Q2884639)

Given a convex body \(K \subset {\mathbb R}^n \) and \( \delta > 0 \), the authors recall the definition of its floating body \( K_{\delta}\) as intersection of all halfspaces \({H}^+\) whose defining hyperplanes \( H \) cut off a set of volume at most \( \delta\) from \( K \), that means \(K_{\delta} = \bigcap_{|H^- \cap K| \leq \delta} H^+ \), and the corresponding Homothety Conjecture: Does \(K\) have to be an ellipsoid if \(K\) is homothetic to \( K_{\delta}\) for some \( \delta > 0\)?NEWLINENEWLINEThe main results of the paper are:NEWLINENEWLINETheorem 3.1. Let \( B^n_p\), \( 1\leq p \leq \infty \) be the unit ball in the \( n\)-dimensional space \( l^n_p\). Let \( 0<\delta < |B^n_p|/2\). Then \( (B^n_p)_{\delta} = c B^n_p\) for some \( 0<c<1\) if and only if \(p = 2 \).NEWLINENEWLINETheorem 4.1. Let \( K \) be a convex body in \( {\mathbb R}^n\). Then there exists a positive number \( \delta (K)\), such that the following are equivalent:NEWLINENEWLINE{(i)} \( K_{\delta}\) is homothetic to \(K\) for some \(0<\delta<\delta(K) \);NEWLINENEWLINE{(ii)} \( K\) is an ellipsoid.NEWLINENEWLINEThe authors give an estimate for such a threshold \( \delta (K)\).NEWLINENEWLINELet \(\{ K_s\}_{s \geq 0} \) be a family of convex bodies constructed from \( K\). Besides the floating bodies \( K_{\delta}\), well-known examples of such families are:NEWLINENEWLINE(1) Illuminating bodies \( K^{\delta} = \{x \in {\mathbb R}^n: |conv (x,K)| - |K| \leq \delta\)\};NEWLINENEWLINE(2) The convolution bodies \( C(K,t) = \{x/2\in {\mathbb R}^n: |K \cap (K+x)|\geq 2t\}\), defined for a symmetric convex body in \({\mathbb R}^n\) and \( t \geq 0\);NEWLINENEWLINE(3) The Santaló regions \( S(R,t)=\{ x \in K: |K^x| \leq 1/t\}\), where \( K^x\) is the polar body of \(K \) with respect to \(x\).NEWLINENEWLINEThe authors formulate:NEWLINENEWLINE Generalized Homothety Conjecture. Let \( K \) be a convex body in \({\mathbb R}^n\). Does \( K_s = cK\) for some \( 0<s\) and some \( c>0\) imply that \(K\) is an ellipsoid?NEWLINENEWLINEUsing the proof of Theorem 4.1, the authors show that in the case of convex bodies with \( C^2\)-boundaries with strictly positive Gauss curvature, the Generalized Homothety Conjecture is valid for all the families (1), (2), (3) listed above.