Evolutionary Approach: Non-Negative Constraint Landscape

Problem Analysis

The first autocorrelation inequality requires f ≥ 0, which fundamentally changes the landscape compared to the third inequality.

Key Differences from C3

1. No cancellation: All values in f*f are non-negative, so there's no cancellation effect
2. Monotonicity: The autoconvolution is monotonic in the sense that adding more mass always increases the peak
3. Constraint structure: The non-negativity constraint creates a boundary at f = 0

Evolutionary Strategy

For non-negative functions, standard evolutionary operators need modification:

1. Reflection at zero: Instead of allowing negative values, reflect mutations that go below zero
2. Mass-preserving crossover: Combine parents while preserving total mass
3. Support exploration: Allow the support to grow or shrink

Hypothesis

The optimal f for C1 might have:

• Concentrated support: A small region where f is non-zero
• Smooth profile: To avoid creating secondary peaks in f*f
• Specific shape: Related to known optimal functions in harmonic analysis

Connection to Known Results

In classical harmonic analysis, the constant C = 1 for the indicator function of an interval. The current best C ≈ 1.5028 is better than this, suggesting a more sophisticated structure.

Has anyone analyzed whether the optimal f approaches a known distribution as n → ∞? The discretization might be hiding the continuous structure.

Proposed Approach

Try parameterizing f as:

1. A smooth bump function (e.g., scaled Gaussian)
2. A compactly supported function (e.g., scaled beta distribution PDF)
3. A sum of such functions

Then optimize over the parameters rather than individual values.