In this paper, we consider the family of nth degree polynomials whose coefficients form a log-convex sequence (up to binomial weights), and investigate their roots. We study, among others, the structure of the set of roots of such polynomials, showing that it is a closed convex cone in the upper half-plane, which covers its interior when n tends to infinity, and giving its precise description for every $n\in \mathbb {N}$ , $n\geq 2$ . Dual Steiner polynomials of star bodies are a particular case of them, and so we derive, as a consequence, further properties for their roots.

We shall prove a convergence result relative to sequences of Minkowski symmetrals of general compact sets. In particular, we investigate the case when this process is induced by sequences of subspaces whose elements belong to a finite family, following the path marked by Klain in Klain (2012, Advances in Applied Mathematics 48, 340–353), and the generalizations in Bianchi et al. (2019, Convergence of symmetrization processes, preprint) and Bianchi et al. (2012, Indiana University Mathematics Journal 61, 1695–1710). We prove an analogous result for fiber symmetrization of a specific class of compact sets. The idempotency for symmetrizations of this family of sets is investigated, leading to a simple generalization of a result from Klartag (2004, Geometric and Functional Analysis 14, 1322–1338) regarding the approximation of a ball through a finite number of symmetrizations, and generalizing an approximation result in Fradelizi, Làngi and Zvavitch (2020, Volume of the Minkowski sums of star-shaped sets, preprint).

In response to an open problem raised by S. Rabinowitz, we prove that

$$ \begin{align*} \begin{array} [c]{l} \left( \left( x^{2}+y^{2}\right) {}^{2}+8y\left( y^{2}-3x^{2}\right) \right) {}^{2}+432y\left( y^{2}-3x^{2}\right) \left( 351-10\left( x^{2}+y^{2}\right) \right) \\ =567^{3}+28\left( x^{2}+y^{2}\right) {}^{3}+486\left( x^{2}+y^{2}\right) \left( 67\left( x^{2}+y^{2}\right) -567\times18\right) \end{array} \end{align*} $$

In the Euclidean plane ${{\mathbb{R}}^{2}}$, we define the Minkowski difference $\mathcal{K}-\mathcal{L}$ of two arbitrary convex bodies $\mathcal{K},\mathcal{L}$ as a rectifiable closed curve ${{\mathcal{H}}_{h}}\subset {{\mathbb{R}}^{2}}$ that is determined by the difference $h={{h}_{K}}-{{h}_{\mathcal{L}}}$ of their support functions. This curve ${{\mathcal{H}}_{h}}$ is called the hedgehog with support function $h$. More generally, the object of hedgehog theory is to study the Brunn–Minkowski theory in the vector space of Minkowski differences of arbitrary convex bodies of Euclidean space ${{\mathbb{R}}^{n+1}}$, defined as (possibly singular and self-intersecting) hypersurfaces of ${{\mathbb{R}}^{n+1}}$. Hedgehog theory is useful for: (i) studying convex bodies by splitting them into a sum in order to reveal their structure; (ii) converting analytical problems into geometrical ones by considering certain real functions as support functions. The purpose of this paper is to give a detailed study of plane hedgehogs, which constitute the basis of the theory. In particular: (i) we study their length measures and solve the extension of the Christoffel–Minkowski problem to plane hedgehogs; (ii) we characterize support functions of plane convex bodies among support functions of plane hedgehogs and support functions of plane hedgehogs among continuous functions; (iii) we study the mixed area of hedgehogs in ${{\mathbb{R}}^{2}}$ and give an extension of the classical Minkowski inequality (and thus of the isoperimetric inequality) to hedgehogs.

We extend the recent results of R. Latała and O. Guédon about equivalence of ${{L}_{q}}$ -norms of logconcave random variables (Kahane-Khinchin’s inequality) to the quasi-convex case. We construct examples of quasi-convex bodies ${{K}_{n\,}}\subset {{\mathbb{R}}^{n}}$ which demonstrate that this equivalence fails for uniformly distributed vector on ${{K}_{n}}$ (recall that the uniformly distributed vector on a convex body is logconcave). Our examples also show the lack of the exponential decay of the “tail” volume (for convex bodies such decay was proved by M. Gromov and V. Milman).

In the original view-obstruction problem congruent closed, centrally symmetric convex bodies centred at the points of the set are expanded uniformly until they block all rays from the origin into the open positive cone. The central problem is to determine the minimal blocking size and this value is known for balls in dimensions n = 2,3 and for symmetrically placed cubes in dimensions n = 2, 3, 4

In order to explain fully the distinction between rational and irrational rays in the original problem, we extend consideration to the blocking of subspaces of all dimensions. In order to appreciate the special properties of balls and cubes, we give a discussion of the convex body with respect to reflection symmetry, lower dimensional sections, and duality. We introduce topological considerations to help understand when the critical parameter of the theory is an attained maximum and we add substantially to the list of known values of this parameter. In particular, when the dimension is n = 2 our dual body considerations furnish a complete solution to the view-obstruction problem