The isotropy constant and boundary properties of convex bodies (Q2814413)

For any convex body \(K\) in \(\mathbb{R}^n\) endowed with its canonical scalar product \(\langle \cdot,\cdot \rangle\) and Euclidean norm \(|\cdot|\), there exists a unique (up to orthogonal transformations) affine, volume preserving, mapping \(A : \mathbb{R}^n \rightarrow \mathbb{R}^n\) such that for some constant \(M_K > 0\), depending on \(K\), one has for every \(y \in \mathbb{R}^n\) NEWLINE\[NEWLINE \int_{AK} \langle x, y \rangle dx = 0 \quad \text{and}\quad \int_{AK} \langle x, y \rangle ^2 dx = M_K^2 |y|^2. NEWLINE\]NEWLINE Then \(K\) is called isotropic if \(A\) is the identity on \( \mathbb{R}^n\). The isotropy constant \(L_K\) of \(K\) is defined by \(L_K =M_K/ |K|^{\frac{n+2}{2n}}\), where \(|K|\) is the the volume of \(K\). NEWLINENEWLINEThe main result of the paper is the following: If a convex body \(K\) in \(\mathbb{R}^n\) is a local maximizer for \(L_K\), then it has no positive generalized Gauss curvature at any point of its boundary; the same is true for a centrally symmetric \(K\) which is a local maximizer for \(L_K\) among centrally symmetric convex bodies. This result generalizes some results from the paper [\textit{S. Campi} et al., Rend. Ist. Mat. Univ. Trieste 31, No. 1--2, 79--94 (1999; Zbl 0969.52001)].