Calculus of variations: the Gaussian has the most entropy

Every density here has the same mean and the same spread as the Gaussian. Entropy is the number the Gaussian wins.

Gaussian against another density with the same mean and spread

Mean μ: 0.0

Std dev σ: 1.0

Alternative density

Mix toward alternative, t: 0.50

Gaussian Alternative (same μ, σ) Mixture (1−t)·Gaussian + t·alternative

H[Gaussian]

H[alternative]

H[mixture]

Gap to maximum

Entropy along the path from the Gaussian to the alternative

The mixture keeps the same mean and variance at every t, so every point on the path obeys the constraints. The entropy peaks at t = 0.

The shortest path on a sphere

Two points on a sphere. The bold orange arc is the great-circle arc. Push it sideways with the slider: every bump makes the path longer, which is the nudge-and-look-at-the-slope argument.

Point A latitude: 20°

Point A longitude: -60°

Point B latitude: 35°

Point B longitude: 70°

Sideways bump ε: 0.00

View: turn the sphere 0°

Great-circle arc (short way) Nudged path The rest of the great circle

Great-circle arc (in radii R)

Nudged path (in radii R)

Extra length