arXiv subjects

Ralph Howard

Publications and source records attributed to Ralph Howard.

18 recordsLinked to original sources

arXiv2025

Continuous nowhere differentiable multivariate functions

Let $U$ be an open set in $\mathbb{R}^d$. A continuous function $f\colon U \to \mathbb{R}$ is strongly nowhere differentiable if and only if for each $\gamma\in(0,1]$ and for each unit speed $C^{1,\gamma}$ curve $c\colon [a,b] \to U$, the composition $f\circ c \colon [a,b] \to \mathbb{R}$ is nowhere differentiable on $(a,b)$. For bounded $U$, let $\overline U$ be the closure of $U$ and $C(\overline U)$ be the Banach space of continuous real-valued functions on $\overline U$ with the sup norm. Theorem. In the sense of the Baire category theorem, almost every $f\in C(\overline U)$ is strongly nowhere differentiable on $U$.

math.CA↗

arXiv2025

The Gronwall inequality

We prove the following version generalization of the Gronwall inequality: Let $\mathbf X$ be a Banach space and $U\subset \mathbf X$ an open convex set in $\mathbf X$. Let $f,g\colon [a,b]\times U\to \mathbf X$ be continuous functions and let $y,z\colon [a,b]\to U$ satisfy the initial value problems \begin{align*} y'(t)&=f(t,y(t)),\quad y(a)=y_0,\\ z'(t)&=g(t,z(t)),\quad z(a)=z_0. \end{align*} Also assume there is a constant $C\ge 0$ so that $$ \|g(t,x_2)-g(t,x_1)\|\le C\|x_2-x_1\| $$ and a continuous function $\phi\colon [a,b]\to [0,\infty)$ so that $$ \|f(t,y(t))-g(t,y(t))\|\le \phi(t). $$ Then for $t\in [a,b]$ $$ \|y(t)-z(t)\| \le e^{C|t-a|}\|y_0-z_0\|+e^{C|t-a|}\int_a^te^{-C|s-a|}\phi(s)\,ds. $$

math.FA↗

arXiv2023

Shur type comparison theorems for affine curves with application to lattice point estimates

If $c, \overline c\colon [a,b]\to \mathbb R^2$ are two convex planar curve parameterized by affine arc length and $A\colon [a,b]\to [0,\infty)$ is the area bounded by the restriction $c\big|_{[a,s]}$ and the segment between $c(a)$ and $c(s)$ with $\overline A$ the corresponding function for $\overline c$, and the affine curvature of $c$ pointwise less than the affine curvature of $\overline c$. Then $A(s)\ge \overline A(s)$. Also for any point of a convex curve we define \emph{adapted affine coordinates} centered at the point and give sharp estimates on the coordinates of the curve in terms of bounds on the curvature. Proving these bounds involves generalizing classical comparison theorems of Strum-Liouville type to higher order and nonhomogenous equations. These estimates allow us to give sharp bounds on the areas of inscribed triangles in terms of affine curvature and the affine distance between the vertices. These inequalities imply upper bounds of the number of lattice points on a convex curve in terms of its affine arc length.

math.DG↗

arXiv2022

Bounding the number of lattice points near a convex curve by curvature

We prove explicit bounds on the number of lattice points on or near a convex curve in terms of geometric invariants such as length, curvature, and affine arclength. In several of our results we obtain the best possible constants. Our estimates hold for lattices more general than the usual lattice of integral points in the plane.

math.NT↗

arXiv2013

Total diameter and area of closed submanifolds

The total diameter of a closed planar curve $C\subset R^2$ is the integral of its antipodal chord lengths. We show that this quantity is bounded below by twice the area of $C$. Furthermore, when $C$ is convex or centrally symmetric, the lower bound is twice as large. Both inequalities are sharp and the equality holds in the convex case only when $C$ is a circle. We also generalize these results to $m$ dimensional submanifolds of $R^n$, where the "area" will be defined in terms of the mod $2$ winding numbers of the submanifold about the $n-m-1$ dimensional affine subspaces of $R^n$.

math.DG↗

arXiv2011

Normal curvatures of asymptotically constant graphs and Caratheodory's conjecture

We show that Caratheodory's conjecture, on umbilical points of closed convex surfaces, may be reformulated in terms of the existence of at least one umbilic in the graphs of functions f: R^2-->R whose gradient decays uniformly faster than 1/r. The divergence theorem then yields a pair of integral equations for the normal curvatures of these graphs, which establish some weaker forms of the conjecture. In particular, we show that there are uncountably many principal lines in the graph of f whose projection into R^2 are parallel to any given direction.

math.DG↗

arXiv2010

Tangent cones and regularity of real hypersurfaces

We characterize embedded $\C^1$ hypersurfaces of $\R^n$ as the only locally closed sets with continuously varying flat tangent cones whose measure-theoretic-multiplicity is at most $m<3/2$. It follows then that any (topological) hypersurface which has flat tangent cones and is supported everywhere by balls of uniform radius is $\C^1$. In the real analytic case the same conclusion holds under the weakened hypothesis that each tangent cone be a hypersurface. In particular, any convex real analytic hypersurface $X\subset\R^n$ is $\C^1$. Furthermore, if $X$ is real algebraic, strictly convex, and unbounded then its projective closure is a $\C^1$ hypersurface as well, which shows that $X$ is the graph of a function defined over an entire hyperplane.

math.AG↗

arXiv2006

Nakajima's problem: convex bodies of constant width and constant brightness

For a convex body $K\subset\R^n$, the $k$th projection function of $K$ assigns to any $k$-dimensional linear subspace of $\R^n$ the $k$-volume of the orthogonal projection of $K$ to that subspace. Let $K$ and $K_0$ be convex bodies in $\R^n$, and let $K_0$ be centrally symmetric and satisfy a weak regularity and curvature condition (which includes all $K_0$ with $\f K_0$ of class $C^2$ with positive radii of curvature). Assume that $K$ and $K_0$ have proportional 1st projection functions (i.e., width functions) and proportional $k$th projection functions. For $2\le k<(n+1)/2$ and for $k=3, n=5$ we show that $K$ and $K_0$ are homothetic. In the special case where $K_0$ is a Euclidean ball, we thus obtain characterizations of Euclidean balls as convex bodies of constant width and constant $k$-brightness.

math.MG↗

arXiv2004

Smooth convex Bodies with proportional projection functions

For a convex body $K\subset\R^n$ and $i\in\{1,...,n-1\}$, the function assigning to any $i$-dimensional subspace $L$ of $\R^n$, the $i$-dimensional volume of the orthogonal projection of $K$ to $L$, is called the $i$-th projection function of $K$. Let $K, K_0\subset \R^n$ be smooth convex bodies of class $C^2_+$, and let $K_0$ be centrally symmetric. Excluding two exceptional cases, that of $(i,j)=(1,n-1)$ and $(i,j)=(n-2,n-1)$, we prove that $K$ and $K_0$ are homothetic if they have two proportional projection functions. The special case when $K_0$ is a Euclidean ball provides an extension of Nakajima's classical three-dimensional characterization of spheres to higher dimensions.

math.MG↗

arXiv2004

A general theory of almost convex functions

Let $Δ_m$ be the standard $m$-dimensional simplex of non-negative $m+1$ tuples that sum to unity and let $S$ be a nonempty subset of $Δ_m$. A real valued function $h$ defined on a convex subset of a real vector space is $S$-almost convex iff for all $(t_0,...,t_m)\in S$ and $x_0,...,x_m\in C$ the inequality h(t_0 x_0+ ... +t_m x_m)\leq 1+ t_0 h(x_0)+ ... +t_m h(x_m) holds. A detailed study of the properties of $S$-almost convex functions is made, including the constriction of the extremal (i.e. pointwise largest bounded) $S$-almost convex function on simplices that vanishes on the vertices. In the special case that $S$ is the barycenter of $Δ_m$ very explicit formulas are given for the extremal function and its maximum. This is of interest as the extremal function and its maximum give the best constants in various geometric and analytic inequalities and theorems.

math.FA↗

arXiv2004

A Generalized Affine Isoperimetric Inequality

A purely analytic proof is given for an inequality that has as a direct consequence the two most important affine isoperimetric inequalities of plane convex geometry: The Blaschke-Santalo inequality and the affine isoperimetric inequality of affine differential geometry.

math.MG↗

arXiv2003

Convex Bodies of Constant Width and Constant Brightness

In 1926 S. Nakajima (= A. Matsumura) showed that any convex body in $\R^3$ with constant width, constant brightness, and boundary of class $C^2$ is a ball. We show that the regularity assumption on the boundary is unnecessary, so that balls are the only convex bodies of constant width and brightness.

math.MG↗

arXiv2001

Circles Minimize most Knot Energies

We define a new class of knot energies (known as renormalization energies) and prove that a broad class of these energies are uniquely minimized by the round circle. Most of O'Hara's knot energies belong to this class. This proves two conjectures of O'Hara and of Freedman, He, and Wang. We also find energies not minimized by a round circle. The proof is based on a theorem of G. Luko on average chord lengths of closed curves.

math.GT↗

arXiv2000

Extremal Approximately Convex Functions and the Best Constants in a Theorem of Hyers and Ulam

Let $n\ge1$ and $B\ge2$. A real-valued function $f$ defined on the $n$-simplex $Δ_n$ is approximately convex with respect to $Δ_{B-1}$ iff f(\sum_{i=1}^B t_ix_i) \le \sum_{i=1}^B t_if(x_i) +1 for all $x_1,...,x_B \in Δ_n$ and all $(t_1,...,t_B)\in Δ_{B-1}$. We determine explicitly the extremal (i.e. pointwise largest) function of this type which vanishes on the vertices of $Δ_n$. We also prove a stability theorem of Hyers-Ulam type which yields as a special case the best constants in the Hyers-Ulam stability theorem for $ε$-convex functions.

math.FA↗

arXiv2000

Regularity of Horizons and The Area Theorem

We prove that the area of sections of future event horizons in space-times satisfying the null energy condition is non-decreasing towards the future under any one of the following circumstances: 1) the horizon is future geodesically complete; 2) the horizon is a black hole event horizon in a globally hyperbolic space-time and there exists a conformal completion with a ``H-regular'' Scri plus; 3) the horizon is a black hole event horizon in a space-time which has a globally hyperbolic conformal completion. (Some related results under less restrictive hypotheses are also established.) This extends a theorem of Hawking, in which piecewise smoothness of the event horizon seems to have been assumed. No assumptions about the cosmological constant or its sign are made. We prove smoothness or analyticity of the relevant part of the event horizon when equality in the area inequality is attained - this has applications to the theory of stationary black holes, as well as to the structure of compact Cauchy horizons. In the course of the proof we establish several new results concerning the differentiability properties of horizons.

gr-qc↗

arXiv2000

Solutions Near Singular Points to the Eikonal and Related First Order Non-linear Partial Differential Equations in Two Independent Variables

A detailed study of solutions to the first order partial differential equation H(x,y,z_x,z_y)=0, with special emphasis on the eikonal equation z_x^2+z_y^2=h(x,y), is made near points where the equation becomes singular in the sense that dH=0, in which case the method of characteristics does not apply. The main results are that there is a strong lack of uniqueness of solutions near such points and that solutions can be less regular than both the function H and the initial data of the problem, but that this loss of regularity only occurs along a pair of curves through the singular point. The main tools are symplectic geometry and the Sternberg normal form for Hamiltonian vector fields.

math.AP↗

arXiv1999

On the size of approximately convex sets in normed spaces

Let X be a normed space. A subset A of X is approximately convex if $d(ta+(1-t)b,A) \le 1$ for all $a,b \in A$ and $t \in [0,1]$ where $d(x,A)$ is the distance of $x$ to $A$. Let $\Co(A)$ be the convex hull and $\diam(A)$ the diameter of $A$. We prove that every $n$-dimensional normed space contains approximately convex sets $A$ with $\mathcal{H}(A,\Co(A))\ge \log_2n-1$ and $\diam(A) \le C\sqrt n(\ln n)^2$, where $\mathcal{H}$ denotes the Hausdorff distance. These estimates are reasonably sharp. For every $D>0$, we construct worst possible approximately convex sets in $C[0,1]$ such that $\mathcal{H}(A,\Co(A))=\diam(A)=D$. Several results pertaining to the Hyers-Ulam stability theorem are also proved.

math.FA↗

arXiv1998

Extremal Approximately Convex Functions and Estimating the Size of Convex Hulls

A real valued function $f$ defined on a convex $K$ is anemconvex function iff it satisfies $$ f((x+y)/2) \le (f(x)+f(y))/2 + 1. $$ A thorough study of approximately convex functions is made. The principal results are a sharp universal upper bound for lower semi-continuous approximately convex functions that vanish on the vertices of a simplex and an explicit description of the unique largest bounded approximately convex function~$E$ vanishing on the vertices of a simplex. A set $A$ in a normed space is an approximately convex set iff for all $a,b\in A$ the distance of the midpoint $(a+b)/2$ to $A$ is $\le 1$. The bounds on approximately convex functions are used to show that in $\R^n$ with the Euclidean norm, for any approximately convex set $A$, any point $z$ of the convex hull of $A$ is at a distance of at most $[\log_2(n-1)]+1+(n-1)/2^{[\log_2(n-1)]}$ from $A$. Examples are given to show this is the sharp bound. Bounds for general norms on $R^n$ are also given.

math.MG↗