mathematics

Shephard's problem

Shephard's problem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Shephard's problem rather than just read about it. In short: In mathematics, Shephard's problem, is the following geometrical question asked by Geoffrey Colin Shephard in 1964: if K and L are centrally symmetric convex bodies in n-dimensional Euclidean space such that whenever K and L are projected onto a hyperplane, the volume of the projection of K is smaller than the volume of the projection of L, then does it follow that the volume of K is smaller than that of L? In this…

Key takeaways

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

Reference excerpt

In mathematics, Shephard's problem, is the following geometrical question asked by Geoffrey Colin Shephard in 1964: if K and L are centrally symmetric convex bodies in n-dimensional Euclidean space such that whenever K and L are projected onto a hyperplane, the volume of the projection of K is smaller than the volume of the projection of L, then does it follow that the volume of K is smaller than that of L? In this case, "centrally symmetric" means that the reflection of K in the origin, −K, is a translate of K, and similarly for L. If πk : Rn → Πk is a projection of Rn onto some k-dimensional hyperplane Πk (not necessarily a coordinate hyperplane) and Vk denotes k-dimensional volume, Shephard's problem is to determine the truth or falsity of the implication

V k ( π k ( K ) ) ≤ V k ( π k ( L ) ) for all 1 ≤ k < n ⟹ V n ( K ) ≤ V n ( L ) . {\displaystyle V_{k}(\pi _{k}(K))\leq V_{k}(\pi _{k}(L)){\mbox{ for all }}1\leq k<n\implies V_{n}(K)\leq V_{n}(L).}

Vk(πk(K)) is sometimes known as the brightness of K and the function Vk o πk as a (k-dimensional) brightness function. In dimensions n = 1 and 2, the answer to Shephard's problem is "yes". In 1967, however, Petty and Schneider showed that the answer is "no" for every n ≥ 3. The solution of Shephard's problem requires Minkowski's first inequality for convex bodies and the notion of projection bodies of convex bodies.

Worked examples

Example 1 — a first encounter with Shephard's problem

Start with the simplest possible case. Write down what Shephard's problem 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 Shephard's problem 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 Shephard's problem 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 Shephard's problem

In research: Shephard's problem 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 Shephard's problem 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: Shephard's problem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Convex analysis, Convex geometry, Geometry problems, so understanding it makes those chapters shorter.
In everyday life: Look for Shephard's problem 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.

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How to study Shephard's problem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Shephard's problem 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 Shephard's problem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Shephard's problem in simple terms?

Why does Shephard's problem 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 Shephard's problem?

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 Shephard's problem.