The mechanisms for the origin and maintenance of biological diversity are not fully understood. It is known that frequency-dependent selection, generating advantages for rare types, can maintain genetic variation and lead to speciation, but in models with simple phenotypes (that is, low-dimensional phenotype spaces), frequency dependence needs to be strong to generate diversity. However, we show that if the ecological properties of an organism are determined by multiple traits with complex interactions, the conditions needed for frequency-dependent selection to generate diversity are relaxed to the point where they are easily satisfied in high-dimensional phenotype spaces. Mathematically, this phenomenon is reflected in properties of eigenvalues of quadratic forms. Because all living organisms have at least hundreds of phenotypes, this casts the potential importance of frequency dependence for the origin and maintenance of diversity in a new light.