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Thirteen Results in Magnitude Theory

  • Venkata Siddharth Pendyala
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Abstract

We prove thirteen results concerning three aspects of magnitude theory. The first four concern small-scale magnitude and microscopic weightings. We identify the derivative of the magnitude function at the origin with maximal energy, allowing the value +∞, and construct finite spaces for which small-scale magnitude diverges or for which the one-point property and finite concentration do not produce a microscopic weighting. A fifth result disproves the conjectured monotonicity of magnitude by giving a fourteen-point metric space of strict negative type whose magnitude function decreases. The remaining eight results concern continuity and dimension. Magnitude is Hausdorff-continuous at finite subsets of finite-dimensional positive definite normed spaces, but need not be continuous on compact subsets, even under convexity, geodesicity, bounded tractability, or simultaneous Hausdorff and Lipschitz convergence. The same constructions disprove an ℓ1-intrinsic-volume formula and two tractability conjectures. Finally, we construct a countable compact space of negative type whose Minkowski dimension is zero and whose lower and upper magnitude dimensions are both infinite.

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Cite

BibTeXRIS
BibTeX
@misc{pendyala2026thirteen,
  title = {{Thirteen Results in Magnitude Theory}},
  author = {Venkata Siddharth Pendyala},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13958},
  eprint = {2610.13958},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13958},
}
RIS
TY  - UNPB
TI  - Thirteen Results in Magnitude Theory
AU  - Venkata Siddharth Pendyala
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.13958
UR  - https://arxiv.org/abs/2610.13958
AB  - We prove thirteen results concerning three aspects of magnitude theory. The first four concern small-scale magnitude and microscopic weightings. We identify the derivative of the magnitude function at the origin with maximal energy, allowing the value +∞, and construct finite spaces for which small-scale magnitude diverges or for which the one-point property and finite concentration do not produce a microscopic weighting. A fifth result disproves the conjectured monotonicity of magnitude by giving a fourteen-point metric space of strict negative type whose magnitude function decreases. The remaining eight results concern continuity and dimension. Magnitude is Hausdorff-continuous at finite subsets of finite-dimensional positive definite normed spaces, but need not be continuous on compact subsets, even under convexity, geodesicity, bounded tractability, or simultaneous Hausdorff and Lipschitz convergence. The same constructions disprove an ℓ_1-intrinsic-volume formula and two tractability conjectures. Finally, we construct a countable compact space of negative type whose Minkowski dimension is zero and whose lower and upper magnitude dimensions are both infinite.
ER  -