Related papers: Planar radial mean bodies are convex

In 1998, R. Gardner and G. Zhang introduced the radial $p$th mean bodies $R_p K$ of a convex body $K$ in $\mathbb{R}^n$ for $p>-1$, which now play an important role in geometric tomography. In this work, we study the Fourier transforms of…

In this work, we extend a classical theorem of Keith Ball on integrals of log-concave functions along rays against the weight $r^{p-1}$ to the previously inaccessible regime $p\in (-1,0)$: if $g:\mathbb R^n\to\mathbb R_+$ is an integrable…

A convexity point of a convex body is a point with the property that the union of the body and its reflection in the point is convex. It is proved that in the plane a typical convex body (in the sense of Baire category) has infinitely many…

If K' and K are convex bodies of the plane such that K' is a subset of K then the perimeter of K' is not greater than the perimeter of K. We obtain the following generalization of this fact. Let K be a convex compact body of the plane with…

In light of the log-Brunn-Minkowski conjecture, various attempts have been made to define the geometric mean of convex bodies. Many of these constructions are fairly complex and/or fail to satisfy some natural properties one would expect of…

In this paper, we study a reverse isoperimetric inequality for planar convex bodies whose radius of curvature is between two positive numbers 0 < $\alpha$ < $\beta$, called ($\alpha$, $\beta$)--convex bodies. We show that among planar…

It is proved that every convex body in the plane has a point such that the union of the body and its image under reflection in the point is convex. If the body is not centrally symmetric, then it has, in fact, three affinely independent…

In this work we present a theorem regarding two convex bodies $K_1, K_2\subset \mathbb{R}^{n}$, $n\geq 3$, and two families of sections of them, given by two families of tangent planes of two spheres $S_i\subset \textrm{int}\textrm{ } K_i$,…

We show that every planar convex body is contained in a quadrangle whose area is less than $(1 - 2.6 \cdot 10^{-7}) \sqrt{2}$ times the area of the original convex body, improving the best known upper bound by W. Kuperberg.

Suppose that $K \subseteq \RR^d$ is a 0-symmetric convex body which defines the usual norm $$ \Norm{x}_K = \sup\Set{t\ge 0: x \notin tK} $$ on $\RR^d$. Let also $A\subseteq\RR^d$ be a measurable set of positive upper density $\rho$. We show…

We show that, for any prime power p^k and any convex body K (i.e., a compact convex set with interior) in Rd, there exists a partition of K into p^k convex sets with equal volume and equal surface area. We derive this result from a more…

In this work we prove that if for a pair of convex bodies $K_1, K_2 \subset \mathbb{R}^n$, $n \geq 3$, there exists a hyperplane $H$ and two distinct points $p_1$ and $p_2$ in $\mathbb{R}^n \setminus H$ such that for every $(n-2)$-plane $M…

For a given $\lambda >0$, a convex body in $\mathbb R^n$ is $\lambda$-convex if it is the intersection of (finitely or infinitely many) balls of radius $1/\lambda$. In this note, we show that among all $\lambda$-convex bodies in $\mathbb…

Similarly to the classic notion in $E^d$, a subset of a positive diameter below $\frac{\pi}{2}$ of a hemisphere of the sphere $S^d$ is called complete, provided adding any extra point increases its diameter. Complete sets are convex bodies…

In this work we study upper bounds for the ratio of successive inner and outer radii of a convex body K. This problem was studied by Perel'man and Pukhov and it is a natural generalization of the classical results of Jung and Steinhagen. We…

In 1951, Bang posed the affine plank conjecture, which remains open: If a convex body in $\mathbb{R}^d$ is covered by planks, then the total relative width of the planks is at least one. We prove a lower bound of $2/(1+\sqrt{d})$ for this…

Let $C\subset {\mathbb R}^n$ be a convex body. We introduce two notions of convexity associated to C. A set $K$ is $C$-ball convex if it is the intersection of translates of $C$, or it is either $\emptyset$, or ${\mathbb R}^n$. The $C$-ball…

We prove that there exists an absolute constant $\alpha >1$ with the following property: if $K$ is a convex body in ${\mathbb R}^n$ whose center of mass is at the origin, then a random subset $X\subset K$ of cardinality ${\rm…

In this paper, we show that if $L_p$ Gaussian surface area measure is proportional to the spherical Lebesgue measure, then the corresponding convex body has to be a centered disk when $p\in[0,1)$. Moreover, we investigate $C^0$ estimate of…

For a convex body $K$ in $\mathbb R^n$, the inequalities of Rogers-Shephard and Zhang, written succinctly, are $$\text{vol}_n(DK)\leq \binom{2n}{n} \text{vol}_n(K) \leq \text{vol}_n(n\text{vol}_n(K)\Pi^\circ K).$$ Here, $DK=\{x\in\mathbb…