Explore the constructive challenges surrounding the boundary and border of the numerical range in this 28-minute lecture from the Hausdorff Trimester Program on Constructive Mathematics. Delve into the analysis of convexity in numerical ranges and its implications in constructive settings. Examine key concepts such as the T approximation, extreme points, state space, and boundary complexity. Gain insights into the role of business in this mathematical context and understand the significance of world containment and spectrum pain. Investigate the origins of numerical range theory and its evolution in construction settings.