The rapid growth of interactive Large Language Model serving has made efficient management of dynamic Key-Value cache footprints increasingly important for inference performance. Modern inference systems overwhelmingly rely on time-centric scheduling heuristics, such as Shortest Job First. However, their classical guarantees are rooted in traditional scheduling modeling, failing to capture the highly dynamic, 2D spatio-temporal geometric growth specific to LLM inference mechanisms. To resolve this, we propose the geometry-aware online scheduling by introducing the Smallest Volume First (SVF) algorithm and its highly efficient variant, 1-bit SVF. Via a novel volume-certificate proof, we establish a worst-case approximation ratio for SVF that approaches \textbf{3} in the high-concurrency regime of LLM serving, substantially improving upon the prior best constant of 48. Building upon this core breakthrough, we complete a comprehensive theoretical taxonomy analyzing our algorithms across different traffic scenarios and information availability. Practically, we seamlessly integrate our approach as a plug-and-play layer in vLLM. Extensive evaluations on Llama-3.1 models demonstrate comprehensive performance gains: SVF delivers strong reductions in both average and tail latency, while 1-bit SVF, with merely a single bit of information, achieves competitive throughput and latency. This work establishes a theoretically sound and empirically proven approach for resolving memory-constrained scheduling in modern LLM deployments. To facilitate future research, our code is available at https://github.com/Li-Beverly-Kong/Geometry-Aware-Online-Scheduling.git.

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