Michael Roysdon

Brown University · Institute for Computational and Experimental Research in Mathematics

We develop the framework of $L_p$ operations for functions by introducing two primary new types $L_{p,s}$ summations for $p>0$: the $L_{p,s}$ convolution sum and the $L_{p,s}$ Asplund sum for functions. The first type is defined as the linear summations of functions in terms of the $L_p$ coefficients ($C_{p,\lambda,t}$, $D_{p,\lambda,t}$), the so-c...

We construct the extension of the curvilinear summation for bounded Borel measurable sets to the L p space for multiple power parameterᾱ = (α 1 , · · · , α n+1) when p > 0. Based on this L p,ᾱ-curvilinear summation for sets and concept of compression of sets, the L p,ᾱ-curvilinear-Brunn-Minkowski inequality for bounded Borel measurable sets and its...

For $p\in[0,1]$, an asymptotic formula is proved for the expected $L_p$ surface area of the random convex hull of a fixed number of points sampled uniformly and independently from the Euclidean unit sphere in $\mathbb{R}^n$. This result is an $L_p$ interpolation of the well-known asymptotic formulas of J. S. M\"uller (1990) from stochastic geometry...

The inequalities of Petty and Zhang are affine isoperimetric-type inequalities providing sharp bounds for Vol n−1 n (K)Voln(Π • K), where ΠK is a projection body of a convex body K. In this paper, we present a number of generalizations of Zhang's inequality to the setting of arbitrary measures. In addition, we introduce extensions of the projection...

In 2011 Lutwak, Yang and Zhang extended the definition of the L p-Minkowski combination (p ≥ 1) introduced by Firey in the 1960s from convex bodies containing the origin in their interiors to all measurable subsets in R n , and as a consequence, extended the L p-Brunn-Minkowski inequality to the setting of all measurable sets. In this paper, we pre...

In this paper we address the following question: given a measure μ on Rn, does there exist a constant C>0 such that, for any m-dimensional subspace H⊂Rn and any convex body K⊂Rn, the following sectional Rogers-Shephard type inequality holds:μ((K−K)∩H)≤Csupy∈Rn⁡μ((y−K)∩H)? We show that this inequality is affirmative in the class of measures with rad...

In convex geometry, a classical and very powerful result is the Roger-Shephard inequality which state that, for any convex body $K \subset \R^n$, \[\vol_n(K-K) \leq \binom{2n}{n}\vol_n(K),\] with equality only when $K$ is an $n$-dimensional simplex, where $\vol_n$ denotes the usual volume in $\R^n$. Recently, the Rogers-Shephard inequality has been...

In this paper we prove a series of Rogers–Shephard type inequalities for convex bodies when dealing with measures on the Euclidean space with either radially decreasing densities or quasi-concave densities attaining their maximum at the origin. Functional versions of classical Rogers–Shephard inequalities are also derived as consequences of our app...

In this paper we prove a series of Rogers-Shephard type inequalities for convex bodies when dealing with measures on the Euclidean space with either radially decreasing densities, or quasi-concave densities attaining their maximum at the origin. Functional versions of classical Rogers-Shephard inequalities are also derived as consequences of our ap...