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We prove new versions of the isomorphic Busemann-Petty problem for two different measures and show how these results can be used to recover slicing and distance inequalities. We also prove a sharp upper estimate for the outer volume ratio…

Approximating convex bodies is a fundamental problem in geometry. Given a convex body $K$ in $\mathbb{R}^d$ for a fixed dimension $d$, the objective is to minimize the number of facets of an approximating polytope for a given Hausdorff…

The renormalization group equation describing the evolution of the metric of the non linear sigma models poses some nice mathematical problems involving functional analysis, differential geometry and numerical analysis. We describe the…

We consider a non-local isoperimetric problem with a repulsive Coulombic term. In dimension three this corresponds to the Gamow's famous liquid drop model. We show that whenever the mass is small the ball is the unique minimizer of the…

Let $K$ and $L$ be two convex bodies in $\mathbb R^n$, $n\geq 2$, with $L\subset \text{int}\, K$. We say that $L$ is an equichordal body for $K$ if every chord of $K$ tangent to $L$ has length equal to a given fixed value $\lambda$. J.…

The famous Minkowski inequality provides a sharp lower bound for the mixed volume $V(K,M[n-1])$ of two convex bodies $K,M\subset\mathbb{R}^n$ in terms of powers of the volumes of the individual bodies $K$ and $M$. The special case where $K$…

We study the problem of existence of regions separating a given amount of volume with the least possible perimeter inside a Euclidean cone. Our main result shows that nonexistence for a given volume implies that the isoperimetric profile of…

Complementing our previous results, we give a classification of all isometries (not necessarily surjective) of the metric space consisting of ball-bodies, endowed with the Hausdorff metric. "Ball bodies" are convex bodies which are…

In this paper, we consider the problem of minimizing a difference-of-convex objective over a nonlinear conic constraint, where the cone is closed, convex, pointed and has a nonempty interior. We assume that the support function of a compact…

The isoperimetric quotient of the whole family of inner and outer parallel bodies of a convex body is shown to be decreasing in the parameter of definition of parallel bodies, along with a characterization of those convex bodies for which…

Given a positive lower semi-continuous density $f$ on $\mathbb{R}^2$ the weighted volume $V_f:=f\mathscr{L}^2$ is defined on the $\mathscr{L}^2$-measurable sets in $\mathbb{R}^2$. The $f$-weighted perimeter of a set of finite perimeter $E$…

In this paper we consider metric fillings of convex bodies. We show that convex bodies $C\subset \mathbb{R}^n$ are the unique minimal fillings of their boundary metrics among all integral current spaces. To this end, we also prove that…

The convex body isoperimetric conjecture in the plane asserts that the least perimeter to enclose given area inside a unit disk is greater than inside any other convex set of area $\pi$. In this note we confirm two cases of the conjecture:…

In this note we link symplectic and convex geometry by relating two seemingly different open conjectures: a symplectic isoperimetric-type inequality for convex domains, and Mahler's conjecture on the volume product of centrally symmetric…

For every given $\beta<0$, we study the problem of maximizing the first Robin eigenvalue of the Laplacian $\lambda_\beta(\Omega)$ among convex (not necessarily smooth) sets $\Omega\subset\mathbb{S}^{n}$ with fixed perimeter. In particular,…

We solve a class of isoperimetric problems on $\mathbb{R}^N $ with respect to weights that are powers of the distance to the origin. For instance we show that if $k\in [0,1]$, then among all smooth sets $\Omega$ in $\mathbb{R} ^N$ with…

In this paper we consider the isoperimetric profile of convex cylinders $K\times\mathbb{R}^q$, where $K$ is an $m$-dimensional convex body, and of cylindrically bounded convex sets, i.e, those with a relatively compact orthogonal projection…

We consider the punctured plane with volume density $|x|^\alpha$ and perimeter density $|x|^\beta$. We show that centred balls are uniquely isoperimetric for indices $(\alpha,\beta)$ which satisfy the conditions $\alpha-\beta+1>0$,…

We investigate the asymptotic best approximation of a smooth, strictly convex body $K$ in $\mathbb{R}^d$ by inscribed polytopes with a restricted number of vertices under the intrinsic volume difference. We prove rigidity phenomena in both…

We consider a generalization of the hyperplane problem to arbitrary measures in place of volume and to sections of lower dimensions. We prove this generalization for unconditional convex bodies and for duals of bodies with bounded volume…