arXiv subjects

James W. Roberts

Publications and source records attributed to James W. Roberts.

5 recordsLinked to original sources

arXiv2025

Does the Muller-Lyer illusion induced by a goalkeeper configuration influence soccer penalty kicks?

In soccer penalty kicks, goalkeepers that orient their arms upward compared to downward can be misperceived as being taller - effectively recreating the Muller-Lyer illusion. The present study elaborates on previous research surrounding a potential illusion-induced bias in penalty kicks. Participants were exposed to goalkeeper configurations within a virtual goal including arms-parallel, arms-down, arms-out and arms-up. They separately judged the perceived size of the goalkeeper, and executed penalty kicks. The perceived size was near fully consistent with the intended illusion. Meanwhile, the penalty kicks indicated wider a horizontal position following arms-out, and lower vertical position following arms-up. Likewise, there was no relation between the biases expressed in perception and action. While goalkeepers can elicit a perceptual illusion, this does not extend to influencing the penalty kick itself. Instead, other contextual cues appeared more relevant including the proximity between the goalkeeper and goalposts, and with it, the available space in the goal.

q-bio.NC↗

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↗

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↗

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↗