ℓₚ Unit Balls in ℝ²
Fundamentals of AI/ML · Prof. Parikshit Pareek & Prof. Jitin Singla · IIT Roorkee
The unit ball of the ℓₚ norm is the set of all x with ‖x‖p ≤ 1. As p changes, the shape morphs in a striking way: the diamond (p=1) becomes the circle (p=2), then approaches the square as p→∞. Slide p to watch.
‖x‖p = ( |x1|p + |x2|p )1/p,
‖x‖∞ = max(|x1|, |x2|)
Circle · p = 2 (Euclidean norm) · smooth, rotationally symmetric
Live values
For a point on the boundary at angle 45°:
x = (—, —)
‖x‖p = 1 (by definition of the unit ball)
‖x‖2 = —
‖x‖1 = —
‖x‖∞ = —
Comparison: all three balls
━━ p = 1
━━ p = 2
━━ p = ∞
What to notice during class
- p = 1 corners on the axes. This is why ℓ₁ regularisation (LASSO) gives sparse solutions — the corners are at the axes, where some coordinates are zero.
- Nesting: for all p ≥ 1, the diamond (p=1) is inside the circle (p=2) is inside the square (p=∞). All three contain only the origin if you shrink them to size 0.
- p < 1 not a norm. Drag p below 1: the shape becomes non-convex (the line between two points on it leaves the set). These are called quasi-norms.
- p = ∞ is robust. A point has small ℓ∞-norm iff every coordinate is small — useful for worst-case error bounds.
- Same ball, different geometry. All three norms define the same notion of "zero", but very different notions of "small" — which is why the choice of norm matters in optimization and regularisation.