# Plurisubharmonic envelopes and support functions

Read [the complete lesson](../../AN02-L139.html), or use [the editable source](../../src/plurisubharmonic-envelopes-and-support-functions.md). It contains the full proofs, four worked examples and six exercises with complete solutions.

The support-set figure shows a triangle, its exact supporting line and the envelope of three exponentials. The annulus figure shows the harmonic comparison barrier with its observation point held fixed as the outer radius grows. Exact coordinates, constants, metrics, formulas and proof locators are in [figures/geometry.json](figures/geometry.json). Both figures have PNG, SVG and original one-page vector PDF versions.

Use Python with NumPy, ReportLab and pypdfium2, then run `python figures/render_figures.py` from this directory. The program writes the figures and geometry into its own figures directory. It reads no book pages. Numerical samples illustrate the exact formulas; the lesson supplies the proofs.

Lars Hörmander, The Analysis of Linear Partial Differential Operators II (1983; second revised printing 1990; reprint 2005), section 16.2, Lemmas 16.2.1–16.2.2, printed pp. 314–315, supplies the targets. The exposition, proofs, examples, solutions and diagrams here are original. Exact earlier scalar slab, compact comparison and analytic log-norm proof inputs are linked in the lesson.

Original material uses [CC0 1.0](LICENSE.txt). Referenced software, fonts and course components retain their own terms.