more README

README.md

@@ -18,3 +18,101 @@ fig.add_trace(Cloud(points).scatter())
 fig.show()
 
 ```
+
+## Background
+
+We are solving the convex optimization problem:
+
+\[
+\min_r \quad r
+\]
+
+subject to the constraint that for each point \( p_i \), the Euclidean
+distance from \( p_i \) to the center of the circle is less than or
+equal to the radius \( r \):
+
+\[
+\| p_i - \text{center} \| \leq r, \quad \forall i = 1, 2, \dots, n
+\]
+
+Where:
+
+- \( p_i \) are the points in \( \mathbb{R}^d \).
+- \( \text{center} \) is the center of the circle we are trying to find.
+- \( r \) is the radius of the circle.
+
+The goal is to minimize the radius \( r \) such that all points
+lie inside or on the boundary of the circle.
+
+---
+
+## Interpretation as a Min/Max Problem
+
+The constraint \( \| p_i - \text{center} \| \leq r \) implies that
+the radius \( r \) must be at least as large as the maximum distance
+from the center to any of the points \( p_i \).
+
+If we define the distance from the center to each point as:
+
+\[
+d_i = \| p_i - \text{center} \|
+\]
+
+Then, the radius \( r \) must satisfy:
+
+\[
+r \geq \max_i d_i
+\]
+
+Thus, the optimization problem becomes:
+
+\[
+r = \min \max_i d_i
+\]
+
+This is a **min-max** problem, where we want to
+minimize the maximum distance from the center to any of the points.
+In other words, we are looking for the smallest possible radius \( r \)
+such that the maximum distance from the center to any point is minimized.
+
+---
+
+## Geometric Interpretation
+
+Geometrically, the problem is about finding the **smallest enclosing circle**
+(or ball in higher dimensions) that contains all the given points.
+The center of the circle is positioned in such a way that the radius
+is minimized, but all points still lie inside or on the boundary of the circle.
+
+- **Convexity**: The objective function \( r = \min \max_i d_i \) is
+convex, as the maximum of a set of convex functions is convex.
+Minimizing a convex function over a convex set is a convex optimization problem.
+
+- **Center and Radius**: The solution involves determining
+both the center and the radius of the circle.
+The optimal center minimizes the maximum distance to any of the points,
+and the optimal radius ensures all points are inside
+or on the boundary of the circle.
+
+---
+
+## Relation to Welzl's Algorithm
+
+In 2D, this problem is often solved using **Welzl's algorithm**.
+Welzl's algorithm is a recursive method for finding the
+smallest enclosing circle. It works by:
+
+1. Randomly choosing a subset of points.
+2. Recursively refining the circle so that all points are inside or on its boundary.
+3. Ensuring the radius is minimized.
+
+---
+
+## Conclusion
+
+The problem you're describing is a **min-max** optimization problem
+aimed at finding the smallest enclosing circle (or ball).
+The radius \( r \) is the smallest value that ensures all points
+lie within or on the boundary of the circle, and the goal
+is to minimize the maximum distance from the center to any
+point in the set.