Convex set

There is a homology between a line segment and a convex set. It is helpful to understand the convex set. A line, a line segment, and one sideline has homology to an affine set, a convex set, and a cone. A line is \(\{y|y=\theta_1 x_1 + \theta_2 x_2, \theta_1 + \theta_2 = 1\}\) if \(\theta_1, \theta_2 \in \mathbb{R}\), a line segment is if \(\theta_1, \theta_2 > 0\) and an one side line if any \(\theta_1, \theta_2 < 0\).

A set \(C\) is affine set if $ y C$ and \(\{y|y=\theta_1 x_1 + \theta_2 x_2, \theta_1 + \theta_2 = 1, x_1, x_2 \in C, \theta_1, \theta_2 \in \mathbb{R} \}\). a convex set is if \(\theta_1, \theta_2 > 0\) and a cone if any \(\theta_1, \theta_2 < 0\).

An affine set is a convex set. But all convex set is not an affine set. It looks the convex set has a stronger condition than affine set i.e. positivity of \(\theta\). But in fact, the convex set has a stronger condition on what it should contain. Because an affine set contains more than a convex set, an affine set satisfies the condition to be a convex set.

Jun Kang
Clinical Assistant Professor of Hospital Pathology

My research interests include pathology, oncology and statistics.