James W. Roberts

activity

19982001

collaborators

4 papers

math.FA2001

A general theory of almost convex functions

S. J. Dilworth, Ralph Howard, James W. Roberts

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…

math.FA2000

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

S. J. Dilworth, Ralph Howard, James W. Roberts

Let $n \geq 1$ and $B \geq 2$ . 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…

math.FA1999

On the size of approximately convex sets in normed spaces

S. J. Dilworth, Ralph Howard, James W. Roberts

Let X be a normed space. A subset A of X is approximately convex if $d (t a + (1 - t) b, A) \leq 1$ for all $a, b \in A$ and $t \in [0, 1]$ where $d (x, A)$ is the distance of $x$ to $A$ . Let $…

math.MG1998

Extremal Approximately Convex Functions and Estimating the Size of Convex Hulls

S. J. Dilworth, Ralph Howard, James W. Roberts

A real valued function $f$ defined on a convex $K$ is anemconvex function iff it satisfies $f ((x + y) /2) \leq (f (x) + f (y)) /2 + 1.$ A thorough study of approximately convex function…