On the symmetric difference metric for convex bodies
Some explicit inequalities are proved that relate the Hausdorff metric for convex bodies with the symmetric difference metric and a corresponding surface area deviation measure. Let \(\mathcal K^n\) denote the class of all nonempty compact convex subsets of the Euclidean dimensional space \(\mathbb R^n\); \(\kappa_n\) denotes the volume, and \(\sigma_n\) the surface area of the \(n\)-dimensional unit ball; \(\operatorname {diam}X\) denotes the diameter, \(\operatorname {int}X\) the interior of any \(X\subset\mathbb R^n\) and \(D=\max\{\operatorname {diam}K,\operatorname {diam}L\}.\) For \(K, L \in \mathcal K^n,\) let us denote by \(\delta(K,L), \Delta_v(K,L)\) and \(\Delta_s(K,L),\) the Hausdorff distance, the symmetric difference distance and the symmetric surface area deviation, respectively. Furthermore, if \(\operatorname {int}(K\cap L)\neq\varnothing\) let \(r\) denote the inradius of \(K\cap L.\) Then the following statements hold: \[ \begin{aligned} \Delta_v(K,L)&\leq \frac{2\kappa_n}{2^{1/n-1}}\left(\frac{D}{2}\right)^{n-1}\delta(K,L),\quad n\geq 2,\\ \text{if int\( (K\cap L)\neq\varnothing,\) then} \delta(K,L)&\leq \left(\frac{n}{\kappa_n-1}\right)^{1/n}\left( \frac{D}{r}\right)^{(n-1)/n}\Delta_v(K,L)^{1/n},\quad n \geq 2,\\ \Delta_s(K,L) &\leq \sigma_n\left(\frac{D}{2}\right)^{n-2} \left(\frac{5D}{2r}\right)^{(n-1)/n}\delta(K,L),\quad n\geq 2,\\ \Delta_s(K,L) &\leq \frac{4\pi D}{r}\delta(K,L),\quad n = 2,\\ \delta(K,L) &\leq\begin{cases} \left(\frac{\sqrt{5}+1}{2\kappa_n-1}\right)^{1/(n-1)}\left(\frac{D}{r}\right)^{(n-2)/(n-1)}\Delta_s(K,L)^{1/(n-1)},& n> 2,\\ \left(\frac{\sqrt{2}+1}{2}D\right)^{1/2}\Delta_s(K, L)^{1/2},& n=2.\end{cases} \end{aligned} \] These inequalities cannot be improved in some sense.
scientific article; zbMATH DE number 1428665
scientific article