6 papers
Convexity of Radial Mean Bodies via an Extension of Ball's Bodies
Dylan Langharst
In this work, we extend a classical theorem of Keith Ball on integrals of log-concave functions along rays against the weight to the previously inaccessible regime $p\in…
On the Fourier Mean Bodies of a Convex Body
Dylan Langharst, Auttawich Manui, Artem Zvavitch
In 1998, R. Gardner and G. Zhang introduced the radial th mean bodies of a convex body , , which have since become important objects in geometr…
The Weighted Minkowski Problem
Dylan Langharst, Jiaqian Liu, Shengyu Tang
The Minkowski problem in convex geometry concerns showing that a given Borel measure on the unit sphere is, up to perhaps a constant, some type of surface area measure of a convex…
Some comments on the mth-order Projection Bodies
Dylan Langharst
The celebrated Petty's projection inequality is a sharp upper bound for the volume of the polar projection body of a convex body. Lutwak introduced the concept of mixed projection…
Higher-Order Reverse Isoperimetric Inequalities for Log-concave Functions
Dylan Langharst, Francisco MarÃn Sola, Jacopo Ulivelli
The Rogers-Shephard and Zhang's projection inequalities are two reverse, affine isoperimetric-type inequalities for convex bodies. Following a classical work by Schneider, both ine…
On Moment-Entropy inequalities in the space of matrices
Dylan Langharst
In a series of works, Lutwak, Yang and Zhang established what could be called affine information theory, which is the study of moment-entropy and Fisher-information-type inequaliti…