The authors study the relation between the optimal Sobolev norms, sharp Galiardo-Nirenberg inequalities and the Minkowski problems for convex bodies. It is proved that if \(1\leq p<\infty\) and \(f:{\mathbb R}^n\rightarrow {\mathbb R}\) has \(L^p\) weak derivative, then there exists a unique origin-symmetric convex body \(K=K_pf\) such that for every even continuous, positive function \(\varphi\) of homogeneous degree \(1\), we have \[ \int_{{\mathbb R}^n}\varphi(-\nabla f(x))^p\,dx = \frac{1}{V(K)}\int_{{\mathbb S}^{n-1}}\varphi(u)^pd S_p(K,u), \] where \(S_p(K,\cdot)\) is the \(L^p\) surface area measure of \(K\). Conversely, if \(K\) is an origin-symmetric convex body, then there exists a function \(f\) with \(L^p\) weak derivative such that \(K_p=K\). Moreover, if in addition \(f\in L^q\cap L^r\), \(1\leq p<n\) and \(0<r\leq np/(n-p)\), then \(f\) satisfies the sharp affine inequality \[V(K_{p}f)^{-1/n}\geq c_{p,r,n}\| f\| _q^{1-\alpha}\| f\| _r^{\alpha}\] for some \(q=q(p,r)\) and \(\alpha=\alpha(p,r,n,q)\). It is proved that the left hand side of the last inequality can be replaced by \(V(B_{p}f)^{-1/n}\) with different \(c_{p,r,n}\) constant. Here, \(B_pf\) is the \(L^p\) polar projection body related to the function \(f\).

Reviewer: Leszek Skrzypczak (Poznań)