We prove that the log-Brunn–Minkowski inequality (log-BMI) for the Lebesgue measure in dimension $n$ would imply the log-BMI and, therefore, theB-conjecture for any even log-concave measure in dimension $n$ . As a consequence, we prove the log-BMI and the B-conjecture for any even log-concave measure in the plane. Moreover, we prove that the log-BMI reduces to the following: for each dimension $n$ , there is a density $f_{n}$ , which satisfies an integrability assumption, so that the log-BMI holds for parallelepipeds with parallel facets, for the density $f_{n}$ . As a byproduct of our methods, we study possible log-concavity of the function $t\mapsto |(K+_{p}\cdot ~\text{e}^{t}L)^{\circ }|$ , where $p\geqslant 1$ and $K$ , $L$ are symmetric convex bodies, which we are able to prove in some instances and, as a further application, we confirm the variance conjecture in a special class of convex bodies. Finally, we establish a non-trivial dual form of the log-BMI.

The primary aim of this paper is to determine how two convex bodies are related if their respective normal bundles are equal or, in a certain sense, close to each other. In connection with this problem some pertinent results concerning parallel bodies and the Steiner point are also discussed.

Let K be a convex body and let p0∈int K. Suppose that in every direction we can choose continuously a section of K which is a translated copy of the corresponding parallel section of K through p0. Our main result essentially claims that if all these pairs of sections are different almost everywhere, then K is an ellipsoid.

Let $K$ be a convex body with boundary of class ${\mathcal{C}}^{\geq 4}$. We prove that, for a sufficiently small $\delta$, the floating body $K_{\delta}$ is homothetic to $K$ if and only if $K$ is an ellipsoid. The proof relies on properties of the affine curvature flow.

The covariogram of a convex body $K$ provides the volumes of the intersections of $K$ with all its possible translates. Matheron conjectured in 1986 that this information determines $K$ among all convex bodies, up to certain known ambiguities. It is proved that this is the case if $K\subset{\mathbb R}^2$ is not ${\rm C}^1$, or it is not strictly convex, or its boundary contains two arbitrarily small ${\rm C}^2$ open portions ‘on opposite sides’. Examples are also constructed that show that this conjecture is false in ${\mathbb R}^n$ for any $n\geq 4$.

The Hessian measures of a (semi-)convex function can be introduced as coefficients of a local Steiner formula. The investigation of Hessian measures is continued by the provision of a geometric characterization of the support of these measures. Then the Radon–Nikodym derivative and the absolute continuity of Hessian measures with respect to Lebesgue measure are explored. As special cases of the results, known results for surface area measures of convex bodies are recovered.

For a convex body $K$ in $\R^d$ and $1\le k\le d-1$, let $P_k(K)$ be the Minkowski sum (average) of all orthogonal projections of $K$ onto $k$-dimensional subspaces of $\R^d$. It is known that the operator $P_k$ is injective if $k\,{\geq}\, d/2$, $k\,{=}\,3$ for all $d$, and if $k=2$, $d\ne 14$.

It is shown that $P_{2k}(K)$ determines a convex body $K$ among all centrally symmetric convex bodies and $P_{2k+1}(K)$ determines a convex body $K$ among all bodies of constant width. Corresponding stability results are also given. Furthermore, it is shown that any convex body $K$ is determined by the two sets $P_k(K)$ and $P_{k'}(K)$ if $1<k<k'$. Concerning the range of $P_k$, $1\le k\le d-2$, it is shown that its closure (in the Hausdorff-metric) does not contain any polytopes other than singletons.