The smallest and unexpectedly large bodies made from shadows

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Family 088 · Convex and metric geometry
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-14

1. The question

Look at a solid object from every direction and measure the area of each shadow. Those directional measurements can themselves describe a new convex body, called the projection body. How large or small can this new body be compared with the original?

The normalization is chosen so that stretching or shearing the original object does not change the answer. Consequently the problem is about affine shape, rather than size or orientation. Ellipsoids are the candidates for the smallest normalized projection volume. Simplices were proposed as the largest.

The two manuscripts in family 088 address these opposite ends. M1 claims that ellipsoids are exactly the minimizers in every dimension at least four, completing the dimensions beyond a separately established three-dimensional theorem. M2 gives a simple product construction exceeding the simplex benchmark in dimension twenty. The latter is an additional explicit counterexample: earlier work had already disproved the unrestricted simplex-maximum conjecture.

2. Background and essential definitions

For a compact convex set $L$, its support function is

$$h_L(u)=\max_{x\in L}\langle u,x\rangle.$$

For a unit vector $u$, this records how far the set extends in the direction $u$. A convex body is compact and convex with nonempty interior.

The projection body $\Pi K$ of a convex body $K\subset\mathbb R^n$ is defined by

$$h_{\Pi K}(u)=\operatorname{vol}_{n-1}(\operatorname{proj}_{u^\perp}K).$$

The right side is the volume of the orthogonal shadow on the hyperplane perpendicular to $u$. Opposite directions give the same shadow, so $\Pi K$ is symmetric about the origin even when $K$ is not.

The quantity of interest is

$$R_n(K)=\frac{|\Pi K|}{|K|^{n-1}}.$$

If $K$ is dilated by a factor $a$, each shadow scales by $a^{n-1}$, and the numerator scales by $a^{n(n-1)}$. The denominator scales by the same factor. More generally, the identity

$$\Pi(TK)=|\det T|\,T^{-\mathsf T}\Pi K$$

makes $R_n$ invariant under every invertible linear map. Translation also leaves it unchanged.

Write $\kappa_j$ for the volume of the Euclidean unit ball in dimension $j$. The projection body of the unit ball in $\mathbb R^n$ is a ball of radius $\kappa_{n-1}$, giving

$$R_n(B_2^n)=\kappa_{n-1}^{,n}\kappa_n^{,2-n}.$$

Every ellipsoid has the same value by affine invariance.

3. Development history and previous partial results

Petty developed the projection-body framework and proposed the minimum problem. A related classical theorem also bears his name, but involves the polar of the projection body. Taking a polar changes the functional, so that established inequality does not settle the unpolarized minimum discussed here.

One approach studies repeated projection bodies. A class-reduction inequality shows that applying $\Pi$ does not increase the normalized functional. This directs attention to bodies related by homothety to their second projection body. Saroglou and Zvavitch proved a local minimum theorem under a closeness condition on surface-area densities. Ivaki’s local uniqueness theorem established rigidity near the ball for an associated second-projection equation. Such local results do not rule out a distant global minimizer.

More recently, Chen, Feng, Li, Xi and Xu proved the unrestricted three-dimensional minimum, including ellipsoid equality. Mielke-Sulz’s 2026 manuscript treats bodies of revolution in every dimension at least three. M1 claims arbitrary bodies in the remaining dimensions $n\geq4$, without using the three-dimensional theorem in that proof.

At the upper end, Brannen’s Volumes of projection bodies (1996) proposed that simplices maximize $R_n$. The three-dimensional paper also gives the simplex upper value there. However, Feng, Hu, Liu and Xu’s On the reverse projection inequality (2026, MathSciDoc:2608.23002) gives counterexamples in every dimension at least nine.

M2 explicitly credits that prior negative answer. Its contribution is a direct Cartesian-product witness with an exact computable value, together with an exponential excess over the simplex benchmark in sufficiently large dimensions. It should not be read as the first disproof of Brannen’s conjecture.

4. What the new manuscripts claim

M1 states that every convex body in $\mathbb R^n$, for every integer $n\geq4$, satisfies

$$R_n(K)\geq\kappa_{n-1}^{,n}\kappa_n^{,2-n},$$

with equality if and only if $K$ is an ellipsoid, including arbitrary translations and invertible linear images of the ball. No symmetry, smoothness or proximity-to-a-ball assumption is imposed.

Combined with the external three-dimensional theorem, this gives the claimed sharp minimum for all $n\geq3$. M1 also records consequences for lower-degree projection quantities, affine inequalities involving gradients, and isoperimetry in normed spaces. Those consequences use earlier implications and, when stated in dimension three, the separately proved three-dimensional input.

For the upper problem, let

$$T_d=\operatorname{conv}(0,e_1,\ldots,e_d), \qquad c_d=R_d(T_d)=\frac{(d+1)d^d}{d!}.$$

M2 states that the twenty-dimensional body $K=T_{10}\times T_{10}$ satisfies

$$\frac{R_{20}(K)}{c_{20}} =\frac{22\,355\,476}{22\,020\,096}>1.$$

Thus two ten-dimensional simplices, placed in independent coordinate blocks, have larger normalized projection volume than a single twenty-dimensional simplex.

The underlying identity is multiplicativity for full-dimensional polytopes $A\subset\mathbb R^r$, $B\subset\mathbb R^s$, with $r,s\geq2$:

$$R_{r+s}(A\times B)=R_r(A)R_s(B).$$

Iterating it, M2 claims that some $\lambda>1$ and dimension threshold $N$ exist such that every $n\geq N$ admits a product of simplices $K_n$ with $R_n(K_n)\geq\lambda^n c_n$. The excess can therefore grow exponentially, rather than remaining a small fixed discrepancy.

5. Why the advance matters

Importance: 5/5. The principal reason is the claimed unrestricted minimum theorem. Petty’s projection-volume conjecture is a central affine extremal problem, and its global equality classification would complete a picture previously known through local arguments, special shapes and dimension three.

The two papers also illustrate why an attractive extremal analogy can be only half right. Ellipsoids would indeed minimize this shadow-based functional, but simplices need not maximize it. The upper problem has more room for products and combinatorial structure than the simple candidate suggests.

M2’s exact witness makes that distinction accessible. One does not need to imagine a delicate perturbation of a complicated polytope: a product of two standard simplices already exceeds the proposed value. Its multiplicative mechanism explains why the failure becomes substantial in high dimension.

The family’s score does not assign equal novelty to both components. M2 acknowledges the earlier counterexamples, while M1 makes the broader claimed advance. Neither result determines the true best upper constant or all maximizing bodies.

6. The main idea

Turn shadow geometry into an inequality for measures

M1 associates a positive measure to a normalized convex body through first variation of volume. The measure satisfies a lower bound when integrated against support functions, and its first absolute directional moments represent the projection body.

The desired geometric inequality is then reduced to a lower bound for a two-measure pairing of the form

$$\iint|\langle x,y\rangle|\,d\eta(x)\,d\zeta(y).$$

The measures retain information from the original bodies; they are not arbitrary probability distributions without constraints. This reformulation separates the projection-body application from a more general analytic inequality.

Remove affine directions and control the remaining harmonics

Functions on the sphere can be decomposed into spherical harmonics, the spherical analogues of Fourier modes. Degree-two components reflect changes related to linear coordinates. Higher-degree components record more complicated deviations from a Euclidean shape.

The manuscript claims a strict estimate for powers of arbitrary norms that controls those higher modes, with equality only in the Euclidean case. The degree-two contribution must be handled by choosing coordinates; it does not vanish automatically.

A variational construction selects a norm associated with the measure. To obtain a global affine position, the argument follows these choices even toward singular matrices, where norms become seminorms. A boundary calculation and fixed-point theorem then produce an invertible coordinate change in which the unwanted degree-two component disappears. The strict harmonic estimate and first-variation equality conditions return the ellipsoid classification.

Test products against the proposed maximum

M2 uses a more elementary route. For a polytope, the projection body can be expressed using its facet areas and normals. Facets of a Cartesian product come from one factor at a time. Their contributions split into complementary coordinate blocks, yielding the multiplicative identity for $R_n$.

The projection body of a standard simplex can be represented as a cube plus one segment, allowing its volume to be computed exactly. Comparing the product of two ten-dimensional simplex values with the twenty-dimensional simplex value produces the stated rational number greater than one.

Repeating the ten-dimensional block gives a fixed growth rate per dimension. That rate exceeds the asymptotic exponential rate of the simplex benchmark; an additional simplex factor accommodates dimensions not divisible by ten. This is the mechanism behind the exponential separation.

7. Scope, limitations, and open questions

Both manuscripts concern the volume of $\Pi K$ itself. Replacing it by its polar would invoke a different family of inequalities and different extremizers.

M1’s new theorem starts at dimension four. The combined all-dimensions-at-least-three conclusion credits the independent three-dimensional result. It includes nonsymmetric and nonsmooth bodies, and its equality statement is global.

M2’s explicit example is in dimension twenty. Earlier work gives counterexamples in smaller dimensions, and the product construction is not claimed to locate the first failing dimension. Its asymptotic conclusion holds above some threshold and does not specify the sharp maximum.

Finding that maximum, identifying its extremizers, and determining quantitative stability near the minimum remain separate questions. The digest reports the manuscripts’ statements and explanatory mechanisms without checking their proofs or independently adjudicating novelty.

8. Sources and further reading

M1 contains the measure reduction, strict norm estimate, affine normalization and equality analysis. M2 gives the facet formula, product identity, exact simplex computation and repeated-product consequence.

The linked three-dimensional paper and local uniqueness paper are primary sources for earlier minimum results. Saroglou’s On the shape of a convex body with respect to its second projection body explains the class-reduction viewpoint. M2’s references identify Brannen’s original upper conjecture and Feng–Hu–Liu–Xu’s earlier counterexamples.

Source manuscripts

• M1. OpenAI, Petty’s projection-volume conjecture in dimensions at least four (September 24, 2026).
• M2. OpenAI, A product counterexample to the simplex maximum for projection-body volume (September 24, 2026).

Sources consulted on 14 October 2026 at repository revision 1581d4c. Download the manuscript citations.