On the convolution bodies of \(L_p\)-addition
The Brunn-Minkowski theory and its analogs play enteral roles in convex geometry. \textit{D. Alonso-Gutiérrez} et al. [Adv. Math. 238, 50--69 (2013; Zbl 1281.52005)] revealed the connection between the convolution set and the Minkowski sum, that is, the convolution set \(A+_{1,\theta} B\) of sets \(A,B \subset \mathbb{R}^n\) has the following representation: \[ A+_{1,\theta} B=\{x \in A+_1 B:|A \cap(x-B)| \geq \theta M(A, B)\}, \] for \(\theta \in [0,1]\), whenever \(M(A, B):=\sup _{x \in A+_1 B}|A \cap(x-B)|\) is finite. Here \(A+_1 B\) is the Minkowski sum of \(A\) and \(B\). Now we can see that the Minkowski sum \(A+_1 B=A+_{1,0} B\). They further established the following Brunn-Minkowski inequality for convolution bodies: \begin{align*} \left|K+_{1,\theta} L\right|^{\frac{1}{n}} \geq\left(1-\theta^{\frac{1}{n}}\right)\left(|K|^{\frac{1}{n}}+|L|^{\frac{1}{n}}\right), \end{align*} where \(K,L \subset \mathbb{R}^n\) are arbitrary convex bodies. In the paper under review, the author extends the above results to the \(L_p\) Brunn-Minkowski theory. First, the author recalls the definition of the \(L_p\) Minkowski sum of \(A,B \subset \mathbb{R}^n\) introduced by \textit{E. Lutwak} et al. [Adv. Appl. Math. 48, No. 2, 407--413 (2012; Zbl 1252.52006)]: \[ A+_p B=\left\{(1-t)^{1 / p^{\prime}} x+t^{1 / p^{\prime}} y: x \in A, y \in B,\,0 \leq t \leq 1\right\}, \] where \(1 / p+1 / p^{\prime}=1\). When \(A,B\) are convex bodies containing the origin, it is equivalent to \textit{W. J. Firey}'s classical definition [Math. Scand. 10, 17--24 (1962; Zbl 0188.27303)]. Then the \(L_p\) convolution set is defined as the following: \[ A+_{p, \theta} B=\left\{x \in A+_p B:\left|(1-t)^{1 / p^{\prime}} A \cap\left(x-t^{1 / p^{\prime}} B\right)\right| \geq \theta^p M_p^t(A, B), \,0 \leq t \leq 1\right\} \] for \(\theta \in[0,1]\), whenever \[ M_p^t(A, B):=\sup \left\{\left|(1-t)^{1 / p^{\prime}} A \cap\left(x-t^{1 / p^{\prime}} B\right)\right|: x \in(1-t)^{1 / p^{\prime}} A+t^{1 / p^{\prime}} B\right\}. \] Finally, the author establishes the following \(L_p\) Brunn-Minkowski inequality for \(L_p\) convolution bodies: \begin{align*} \left|K+_{p,\theta} L\right|^{\frac{p}{n}} \geq\left(1-\theta^{\frac{p}{n}}\right)\left(|K|^{\frac{p}{n}}+|L|^{\frac{p}{n}}\right). \end{align*} where \(K,L \subset \mathbb{R}^n\) are convex bodies containing the origin in their interiors, under the assumption that \(M_p^t(K, L)=\left|(1-t)^{1 / p^{\prime}} K \cap\left(-t^{1 / p^{\prime}} L\right)\right|\). Remark that the additional assumption holds if \(K=-L\) or \(K,L\) are origin-symmetric.
scientific article; zbMATH DE number 7468850
scientific article