Brand Space Geometry: A Formal Metric for Multi-Dimensional Brand Perception

Brand theory lacks formal mathematical foundations. While frameworks such as Keller’s Customer-Based Brand Equity and Aaker’s brand personality dimensions provide conceptual taxonomies, none defines a metric space in which brand perception can be rigorously measured, compared, or analyzed. This paper provides the first formal metric space for multi-dimensional brand perception, grounded in Spectral Brand Theory (SBT), which models brands as emitters of signals across eight typed dimensions perceived by observers with heterogeneous salience profiles. We define three metric spaces: the brand signal space (ℬ︀, 𝑑ℬ︀), where brand emission profiles reside in ℝ8+ equipped with the Aitchison metric via isometric log-ratio coordinates; the observer weight space (𝒪︀, 𝑑𝒪︀), where observer salience profiles reside on the probability simplex Δ7 equipped with the Fisher-Rao metric, justified by Cencov’s uniqueness theorem; and the combined brand-observer space (𝒫︀, 𝐷), formalized as a warped product manifold that generalizes the empirically validated INDSCAL model of individual differences scaling. We prove that each space satisfies the metric axioms, derive geodesics, and establish concentration-of-measure bounds at 𝑛 = 8: the expected squared Euclidean distance between uniformly random observer profiles is 7/36, providing a null-model baseline for meaningful brand differentiation. We prove that the positive-octant restriction compresses brand space to 1/256 of the full 7-sphere, quantifying the geometric constraint on brand differentiation. Applications to five SBT case-study brands demonstrate that the metric recovers qualitative coherence assessments and reveals observer-dependent distance structures invisible to traditional brand-comparison methods.