Discrete image tokenization is a key bottleneck for scalable visual generation: a tokenizer must remain compact for efficient latent-space priors while preserving semantic structure and using discrete capacity effectively. Existing quantizers face a trade-off: vector-quantized tokenizers learn flexible geometries but often suffer from biased straight-through optimization, codebook under-utilization, and representation collapse at large vocabularies. Structured scalar or implicit tokenizers ensure stable, near-complete utilization by design, yet rely on fixed discretization geometries that may allocate capacity inefficiently under heterogeneous latent statistics. We introduce Learnable Geometric Quantization (LGQ), a discrete image tokenizer that learns discretization geometry end-to-end. LGQ replaces hard nearest-neighbor lookup with temperature-controlled soft assignments, enabling fully differentiable training while recovering hard assignments at inference. The assignments correspond to posterior responsibilities of an isotropic Gaussian mixture and minimize a variational free-energy objective, provably converging to nearest-neighbor quantization in the low-temperature limit. LGQ combines a token-level peakedness regularizer with a global usage regularizer to encourage confident yet balanced code utilization without imposing rigid grids. Under a controlled VQGAN-style backbone on ImageNet across multiple vocabulary sizes, LGQ achieves stable optimization and balanced utilization. At 16K codebook size, LGQ improves rFID by 11.88% over FSQ while using 49.96% fewer active codes, and improves rFID by 6.06% over SimVQ with 49.45% lower effective representation rate, achieving comparable fidelity with substantially fewer active entries. Our GitHub repository is available at: https://github.com/KurbanIntelligenceLab/LGQ