Holmes–Thompson volume

Key takeaways

Holmes–Thompson volume belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Holmes–Thompson volume to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Holmes–Thompson volume from memory before moving on to harder problems.

Reference excerpt

In geometry of normed spaces, the Holmes–Thompson volume is a notion of volume that allows to compare sets contained in different normed spaces (of the same dimension). It was introduced by Raymond D. Holmes and Anthony Charles Thompson.

Definition The Holmes–Thompson volume Vol HT ⁡ ( A ) {\displaystyle \operatorname {Vol} _{\text{HT}}(A)} of a measurable set A ⊆ R n {\displaystyle A\subseteq R^{n}} in a normed space ( R n , ‖ − ‖ ) {\displaystyle (\mathbb {R} ^{n},\|-\|)} is defined as the 2n-dimensional measure of the product set A × B ∗ , {\displaystyle A\times B^{*},} where B ∗ ⊆ R n {\displaystyle B^{*}\subseteq \mathbb {R} ^{n}} is the dual unit ball of ‖ − ‖ {\displaystyle \|-\|} (the unit ball of the dual norm ‖ − ‖ ∗ {\displaystyle \|-\|^{*}} ).

Symplectic (coordinate-free) definition The Holmes–Thompson volume can be defined without coordinates: if A ⊆ V {\displaystyle A\subseteq V} is a measurable set in an n-dimensional real normed space ( V , ‖ − ‖ ) , {\displaystyle (V,\|-\|),} then its Holmes–Thompson volume is defined as the absolute value of the integral of the volume form 1 n ! ω ∧ ⋯ ∧ ω ⏞ n {\displaystyle {\frac {1}{n!}}\overbrace {\omega \wedge \cdots \wedge \omega } ^{n}} over the set A × B ∗ {\displaystyle A\times B^{*}} ,

Vol H T ⁡ ( A ) = | ∫ A × B ∗ 1 n ! ω n | {\displaystyle \operatorname {Vol} _{HT}(A)=\left|\int _{A\times B^{*}}{\frac {1}{n!}}\omega ^{n}\right|}

where ω {\displaystyle \omega } is the standard symplectic form on the vector space V × V ∗ {\displaystyle V\times V^{*}} and B ∗ ⊆ V ∗ {\displaystyle B^{*}\subseteq V^{*}} is the dual unit ball of ‖ − ‖ {\displaystyle \|-\|} . This definition is consistent with the previous one, because if each point x ∈ V {\displaystyle x\in V} is given linear coordinates ( x i ) 0 ≤ i < n {\displaystyle (x_{i})_{0\leq i<n}} and each covector ξ ∈ V ∗ {\displaystyle \xi \in V^{*}} is given the dual coordinates ( x i i ) 0 ≤ i < n {\displaystyle (xi_{i})_{0\leq i<n}} (so that ξ ( x ) = ∑ i ξ i x i {\displaystyle \xi (x)=\sum _{i}\xi _{i}x_{i}} ), then the standard symplectic form is ω = ∑ i d x i ∧ d ξ i {\displaystyle \omega =\sum _{i}\mathrm {d} x_{i}\wedge \mathrm {d} \xi _{i}} , and the volume form is

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Holmes–Thompson volume

Start with the simplest possible case. Write down what Holmes–Thompson volume claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Holmes–Thompson volume before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Holmes–Thompson volume ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Holmes–Thompson volume

In research: Holmes–Thompson volume appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Holmes–Thompson volume in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Holmes–Thompson volume is common in secondary-school and first-year university syllabi. It links to neighbouring topics Differential geometry, Finsler geometry, Integral geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Holmes–Thompson volume outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Holmes–Thompson volume in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Holmes–Thompson volume means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Holmes–Thompson volume out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Holmes–Thompson volume in simple terms?

In geometry of normed spaces, the Holmes–Thompson volume is a notion of volume that allows to compare sets contained in different normed spaces (of the same dimension). It was introduced by Raymond D.

Why does Holmes–Thompson volume matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Holmes–Thompson volume?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Holmes–Thompson volume.