Dar's conjecture and the log-Brunn-Minkowski inequality (Q291882)

The authors prove the validity of Dar's conjecture for all planar convex bodies, along with a characterization of the equality conditions (in the plane).NEWLINENEWLINELet \(K,L\) be convex bodies in \(\mathbb{R}^n\), i.e., compact and convex sets in \(\mathbb{R}^n\) with non-empty interior. In [Geom. Dedicata 77, No. 1, 1--9 (1999; Zbl 0938.52008)], \textit{S. Dar} conjectured the following interesting strengthening of the Brunn-Minkowski inequality NEWLINE\[NEWLINE V_n(K+L)^{1/n}\geq M(K,L)^{1/n}+\frac{V_n(K)V_n(L)}{M(K,L)}^{1/n} NEWLINE\]NEWLINE for all convex bodies, where \(V_n\) denotes the volume, i.e., the Lebesgue measure and NEWLINE\[NEWLINE M(K,L)=\max_{x\in\mathbb{R}^n} V_n(K\cap (x+L)). NEWLINE\]NEWLINE Dar [loc. cit.] proved the validity of the conjecture in some special cases. However, the conjecture was, so far, open even for planar origin symmetric convex bodies.NEWLINENEWLINEThe authors establish that equality holds (in the plane) if and only if one of the following conditions holds: {\parindent=0.6cm\begin{itemize}\item[(i)] \(K\) and \(L\) are parallelograms with parallel sides, and \(V_2(K)=V_2(L)\). \item[(ii)] \(K\) and \(L\) are homothetic. NEWLINENEWLINE\end{itemize}} For the proof, the authors make use of a special position of the pair of convex bodies \(K,L\), called ``dilation position'' -- which needs not be unique. This position allows them further to study the connections of Dar's conjecture and the log-Brunn-Minkowski inequality [\textit{K. Böröczky jun.} et al., Adv. Math. 231, No. 3--4, 1974--1997 (2012; Zbl 1258.52005)]:NEWLINENEWLINEIf \(K,L\) are origin-symmetric convex bodies in the plane, then for any \(\lambda\in[0,1],\) NEWLINE\[NEWLINE V_n((1-\lambda)\cdot K +_0 \lambda \cdot L)\geq V_n(K)^{1-\lambda}V(L)^{\lambda}, NEWLINE\]NEWLINE where, if \(\lambda\in(0,1)\) NEWLINE\[NEWLINE(1-\lambda)\cdot K+_0 \lambda \cdot L=\bigcap_{ u\in S^{n-1}}\{x\in\mathbb{R}^n: \langle x,u \rangle\leq h_K^{1-\lambda}(u)\, h_L^{\lambda}(u)\},NEWLINE\]NEWLINE \((1-\lambda)\cdot K+_0 \lambda \cdot L=K\) if \(\lambda=0\) and \((1-\lambda)\cdot K+_0 \lambda \cdot L=L\) for \(\lambda=1\). Here \(h_K\) denotes the support function of \(K\).NEWLINENEWLINEAlthough it is known that the log-Brunn-Minkowski inequality cannot hold true for all convex bodies [loc. cit.], the authors prove its validity for all planar convex bodies in dilation position. The equality conditions remain the same as in the symmetric case.