Chebyshev Distance (Maximum norm)

Calculator to compute the L∞-norm with formulas and examples

Chebyshev Distance Calculator

What is calculated?

The Chebyshev distance (also called the maximum norm or L∞-norm) is the maximum of the absolute differences of all components between two vectors.

Input vectors

Values separated by spaces

Same number of values as Vector X

Result
Chebyshev distance (L∞):
Maximum of the absolute differences across components

Chebyshev Info

Properties

Chebyshev distance:

  • Also called the L∞-norm
  • Maximum of component-wise differences
  • Limit case of Minkowski distance (p→∞)
  • Defines the "chessboard distance"

Chessboard: Equals the minimum number of king moves from one square to another on a chessboard.

Special cases
2D space:
Chessboard distance between squares
Unit square:
Distance from center to corners
Maximum component:
Determines the entire distance
Related norms

→ Manhattan distance (L₁)
→ Euclidean distance (L₂)
→ Minkowski distance (Lₚ)

Formulas for Chebyshev distance

Definition
\[d_\infty(x,y) = \max_{i} |x_i - y_i|\] Maximum of component-wise differences
As a limit
\[d_\infty(x,y) = \lim_{p \to \infty} \left(\sum_{i=1}^n |x_i-y_i|^p\right)^{1/p}\] Limit case of the Minkowski distance
Vector norm
\[\|x\|_\infty = \max_{i} |x_i|\] Maximum norm of a vector
Matrix norm
\[\|A\|_\infty = \max_{i} \sum_{j=1}^n |a_{ij}|\] Maximum row sum
2D chessboard
\[d_\infty((x_1,y_1), (x_2,y_2)) = \max(|x_2-x_1|, |y_2-y_1|)\] Chessboard distance
Symmetry
\[d_\infty(x,y) = d_\infty(y,x)\] Symmetric property

Detailed calculation example

Example: compute Chebyshev([0,3,4,5], [7,6,3,-1])

Given:

  • x = [0, 3, 4, 5]
  • y = [7, 6, 3, -1]

Step 1 - Differences:

  • |0 - 7| = 7
  • |3 - 6| = 3
  • |4 - 3| = 1
  • |5 - (-1)| = 6

Step 2 - Determine maximum:

\[d_\infty = \max(7, 3, 1, 6) = 7\]

Interpretation: The largest difference in any single component determines the full distance.

Chessboard example

Example: king moves on the chessboard

Problem:

A king stands on square (1,1) and needs to move to (4,3). How many moves does it need at minimum?

Solution with Chebyshev:

\[d_\infty((1,1), (4,3)) = \max(|4-1|, |3-1|)\] \[= \max(3, 2) = 3\]

One possible move sequence:

(1,1) → (2,2) → (3,3) → (4,3)

Result: The king needs at least 3 moves.

Comparison of Lₚ norms

For vectors [1,2,3] and [4,1,1]
L₁ (Manhattan)
6.000

|3|+|1|+|2| = 6

L₂ (Euclidean)
3.742

√(9+1+4) = √14

L₅ (Minkowski)
3.075

(3⁵+1⁵+2⁵)^(1/5)

L∞ (Chebyshev)
3.000

max(3,1,2) = 3

Observation: As p increases the Lₚ-norm approaches the Chebyshev distance.

Practical applications

Game theory
  • Chessboard navigation
  • Grid-based games
  • Pathfinding algorithms
  • Strategic movement
Robotics
  • Motion planning
  • Collision avoidance
  • Grid navigation
  • Control systems
Numerics
  • Error analysis
  • Convergence criteria
  • Algorithm stability
  • Optimization

Mathematical properties

Norm properties
  • Positivity: ‖x‖∞ ≥ 0, ‖x‖∞ = 0 ⟺ x = 0
  • Homogeneity: ‖αx‖∞ = |α|‖x‖∞
  • Triangle inequality: ‖x+y‖∞ ≤ ‖x‖∞ + ‖y‖∞
  • Equivalence: Equivalent to other norms on finite-dimensional spaces
Geometric properties
  • Unit ball: Hypercube (cube in nD)
  • Convex: Unit ball is convex
  • Polytopal: Unit ball is a polytope
  • Facets: 2ⁿ facets in n dimensions
Important relations

With other norms:
‖x‖∞ ≤ ‖x‖₂ ≤ √n ‖x‖∞
‖x‖∞ ≤ ‖x‖₁ ≤ n ‖x‖∞

Duality:
The Chebyshev norm (L∞) is dual to the L₁ norm (Manhattan)

Geometric interpretation

2D unit balls of different norms
L₁ (Manhattan)
Diamond/Rhombus
♦
L₂ (Euclidean)
Circle
●
L₅ (Minkowski)
Superellipse
⬟
L∞ (Chebyshev)
Square
■

Interpretation: The Chebyshev norm produces a square in 2D and a hypercube in nD as its unit ball.

Distance functions

Bray Curtis Distance  •  Canberra Distance  •  Chebyshev Distance  •  Cosine Similarity  •  Euclidean  •  Levenshtein Distance  •  Manhattan Distance  •  Matching Distance  •  Minkowski Distance  •  Pearson Correlation Coefficient  •