On Blaschke's extension of Bonnesen's inequality
For compact, convex sets \(K_ 1,K_ 2 \subset E^ 2\) with the origin as interior point, let \[ \begin{aligned} r_ i(K_ 1,K_ 2) &= \sup\{r \geq 0: rK_ 2 + x\subseteq K_ 1,\quad x\in K_ 1\},\\R_ u(K_ 1,K_ 2) &= \inf\{r > 0: rK_ 2 + x \supseteq K_ 1,\quad x\in K_ 1\}\tag{1}\end{aligned} \] denote the relative inner and outer radius of \(K_ 1\) w.r.t. \(K_ 2\). Blaschke generalized a result of Bonnesen by showing \[ A(K_ 1) - 2tW(K_ 1,K_ 2) + t^ 2A(K_ 2) \leq 0,\tag{2} \] with \(r_ i(K_ 1,K_ 2) \leq t \leq R_ u(K_ 1,K_ 2)\). (Here \(A\) denotes the area and \(W\) the mixed area of \(K_ 1\) and \(K_ 2\).) In the present paper, (2) is improved for the special case that \(K_ 2 \subset E^ 2\) is a smooth, strictly convex body symmetric about the origin. Namely, for such a \(K_ 2\) there is a unique point \(\widetilde{x} \in K_ 1\) such that the difference of the two quantities in (1), with \(x\) from there replaced by \(\widetilde{x}\), attains its minimum. On that base the authors show \[ A(K_ 1) - 2tW(K_ 1,K_ 2) + t^ 2A(K_ 2) \leq 0 \] for \(r_{\widetilde{x}}(K_ 1,K_ 2) \leq t \leq R_{\widetilde{x}}(K_ 1,K_ 2)\), yielding also \[ [W(K_ 1,K_ 2)]^ 2 - A(K_ 1)\cdot A(K_ 2) = [A(K_ 2)]^ 2\cdot {[R_{\widetilde{x}}(K_ 1,K_ 2)-r_{\widetilde{x}}(K_ 1,K_ 2)]^ 2\over 4}. \]
scientific article; zbMATH DE number 493316