Mixed Christoffel Problem in Convex Geometry

• Mixed Christoffel problem is an inverse problem that seeks a convex body whose mixed area measure matches a given Borel measure.
• It involves solving a linear elliptic PDE combined with geometric rank conditions to ensure the support function represents a C²⁺ convex body.
• Recent advances extend the framework to anisotropic, capillary, and nonlinear curvature flow settings, broadening its practical applications.

Searching arXiv for papers on the mixed Christoffel problem and closely related Christoffel–Minkowski variants. The mixed Christoffel problem is an inverse problem for mixed area measures of convex bodies. In one standard formulation, given $C^{2,+}$ convex bodies $\Omega_1,\dots,\Omega_{n-1}$ and a Borel measure $\mu$ on $\mathbb S^n$, one seeks a convex body $\Omega$ such that

$S(\Omega_1,\dots,\Omega_{n-1},\Omega;\cdot)=\mu.$

In a broader anisotropic form, one asks for necessary and sufficient conditions on $\mu$ so that $\mu=S_i(K,C;\cdot)$ or, in polarized notation, $S(K^{[i]},C_1,\dots,C_{n-i-1};\cdot)$ for some free body $K$. This includes the ordinary Christoffel problem when the fixed bodies are Euclidean balls, and it lies within the wider Christoffel–Minkowski hierarchy of prescribed curvature and area-measure equations (Colesanti et al., 9 Dec 2025, Brauner et al., 13 Aug 2025).

1. Basic objects and compatibility conditions

For a convex body $\Omega_1,\dots,\Omega_{n-1}$0, the support function is

$\Omega_1,\dots,\Omega_{n-1}$1

If $\Omega_1,\dots,\Omega_{n-1}$2 and the Gauss curvature is everywhere positive, the inverse Weingarten form is

$\Omega_1,\dots,\Omega_{n-1}$3

The condition $\Omega_1,\dots,\Omega_{n-1}$4 is equivalent to $\Omega_1,\dots,\Omega_{n-1}$5 being $\Omega_1,\dots,\Omega_{n-1}$6, while $\Omega_1,\dots,\Omega_{n-1}$7 is enough for $\Omega_1,\dots,\Omega_{n-1}$8 to be the support function of a convex body (Colesanti et al., 9 Dec 2025).

Mixed area measures are characterized by the mixed-volume identity

$\Omega_1,\dots,\Omega_{n-1}$9

and in the smooth case they admit the density representation

$\mu$0

where $\mu$1 is the mixed discriminant. In particular, for the $\mu$2-th area measure,

$\mu$3

(Colesanti et al., 9 Dec 2025).

The mixed area measure is nonnegative, centered, symmetric in its arguments, translation invariant in each argument, and local in the sense that it depends only on the corresponding exposed faces. Translation invariance yields the standard compatibility condition

$\mu$4

or, for a smooth density $\mu$5,

$\mu$6

(Brauner et al., 13 Aug 2025, Colesanti et al., 9 Dec 2025).

A useful point of terminology is that current work uses “mixed Christoffel problem” in several adjacent senses. In the narrowest sense it refers to the first mixed area measure $\mu$7; in a broader sense it includes mixed Christoffel–Minkowski problems $\mu$8 and related prescribed-curvature equations built from support functions and elementary symmetric functions of the principal radii.

2. Linear elliptic formulation and the geometricity problem

When $\mu$9 is absolutely continuous and the reference bodies are smooth, the mixed Christoffel problem becomes a linear second-order elliptic equation on the sphere. If $\mathbb S^n$0 is the unknown inverse Weingarten form and

$\mathbb S^n$1

is the mixed cofactor matrix determined by the fixed bodies, then

$\mathbb S^n$2

is equivalent to

$\mathbb S^n$3

with $\mathbb S^n$4 the unknown support function (Colesanti et al., 9 Dec 2025).

This PDE formulation isolates the central analytic difficulty. Solving the linear equation under ellipticity is standard, but a solution $\mathbb S^n$5 of the PDE need not automatically be geometric: one still needs

$\mathbb S^n$6

and ideally $\mathbb S^n$7, in order for $\mathbb S^n$8 to be the support function of a $\mathbb S^n$9 convex body. The mixed Christoffel problem therefore separates into an elliptic solvability problem and a rank-positivity problem for $\Omega$0 (Colesanti et al., 9 Dec 2025).

A recent solution of this issue is based on a constant rank theorem for linear elliptic equations on $\Omega$1. Under a convexity hypothesis on $\Omega$2, any solution with $\Omega$3 has constant rank, and on the whole sphere constant rank $\Omega$4 is excluded; consequently $\Omega$5 everywhere. In the mixed Christoffel application this yields a sufficient existence theorem: if the fixed bodies are $\Omega$6 with positive Gauss curvature, $\Omega$7 is positive and satisfies the compatibility condition, the $\Omega$8-homogeneous extension of $\Omega$9 is convex, and the paper’s matrix convexity condition on $S(\Omega_1,\dots,\Omega_{n-1},\Omega;\cdot)=\mu.$0 holds, then there exists a convex body $S(\Omega_1,\dots,\Omega_{n-1},\Omega;\cdot)=\mu.$1 such that

$S(\Omega_1,\dots,\Omega_{n-1},\Omega;\cdot)=\mu.$2

and the solution is unique up to translation (Colesanti et al., 9 Dec 2025).

This analytic structure is the mixed analogue of the Guan–Ma philosophy for the ordinary Christoffel problem. The coefficient tensor is no longer a scalar multiple of the identity; it is a variable tensor built from the mixed cofactor matrix of the fixed bodies.

3. Complete classifications in special anisotropic geometries

Several recent works obtain necessary-and-sufficient conditions in highly structured anisotropic settings. These results are notable because they solve the inverse problem completely, rather than only under sufficient convexity hypotheses.

Regime	Hypotheses on $S(\Omega_1,\dots,\Omega_{n-1},\Omega;\cdot)=\mu.$3	Conclusion
Disk reference $S(\Omega_1,\dots,\Omega_{n-1},\Omega;\cdot)=\mu.$4	$S(\Omega_1,\dots,\Omega_{n-1},\Omega;\cdot)=\mu.$5 nonnegative, centered, finite, $S(\Omega_1,\dots,\Omega_{n-1},\Omega;\cdot)=\mu.$6, plus disintegration and support-function conditions	Existence of $S(\Omega_1,\dots,\Omega_{n-1},\Omega;\cdot)=\mu.$7; uniqueness up to translation
Bodies of revolution $S(\Omega_1,\dots,\Omega_{n-1},\Omega;\cdot)=\mu.$8	centered zonal $S(\Omega_1,\dots,\Omega_{n-1},\Omega;\cdot)=\mu.$9 with support, positivity, endpoint, and equatorial conditions	Complete classification in the axially symmetric setting
Rotationally symmetric $\mu$0	finite, centered, $\mu$1-invariant $\mu$2, not concentrated on the equator	Existence of a body of revolution $\mu$3 with explicit support-function formula

For the disk area measure, the fixed reference body is an $\mu$4-dimensional disk $\mu$5 contained in a hyperplane orthogonal to an axis $\mu$6. The problem asks when

$\mu$7

for some convex body $\mu$8. The solution proceeds by disintegrating $\mu$9 over the Grassmannian of $\mu=S_i(K,C;\cdot)$0-planes containing $\mu=S_i(K,C;\cdot)$1, reducing the $\mu=S_i(K,C;\cdot)$2-dimensional mixed area-measure problem to a family of planar Christoffel problems. The main theorem states that for a nonnegative, centered, finite Borel measure with $\mu=S_i(K,C;\cdot)$3, existence is equivalent to three conditions: $\mu=S_i(K,C;\cdot)$4 is absolutely continuous with a continuous density; almost every planar disintegration $\mu=S_i(K,C;\cdot)$5 is centered; and there exists a support function $\mu=S_i(K,C;\cdot)$6 satisfying a Berg-type kernel identity on each slice. The solution is unique up to translation. The exclusion of mass at the poles is both a technical condition for disintegration and a geometric condition for uniqueness, because pole masses encode only the mean width in the axis direction (Brauner et al., 13 Aug 2025).

For bodies of revolution, the mixed Christoffel–Minkowski problem is solved without regularity assumptions in the zonal class. For $\mu=S_i(K,C;\cdot)$7, a centered zonal Borel measure $\mu=S_i(K,C;\cdot)$8 on $\mu=S_i(K,C;\cdot)$9 is of the form

$S(K^{[i]},C_1,\dots,C_{n-i-1};\cdot)$0

for a body of revolution $S(K^{[i]},C_1,\dots,C_{n-i-1};\cdot)$1 if and only if five conditions hold: a support restriction determined by the normal cones of the reference body $S(K^{[i]},C_1,\dots,C_{n-i-1};\cdot)$2; non-concentration on the equator; positivity of an induced signed Radon measure involving the profile function $S(K^{[i]},C_1,\dots,C_{n-i-1};\cdot)$3; existence and finiteness of two boundary limits at the poles; and an equatorial mass inequality involving $S(K^{[i]},C_1,\dots,C_{n-i-1};\cdot)$4 and the maximal vertical segment length $S(K^{[i]},C_1,\dots,C_{n-i-1};\cdot)$5. This classification extends and sharpens Firey’s theorem, recovers the isotropic case $S(K^{[i]},C_1,\dots,C_{n-i-1};\cdot)$6, and identifies the uniqueness mechanism: if the admissible support region is all of $S(K^{[i]},C_1,\dots,C_{n-i-1};\cdot)$7, the solution is unique up to vertical translation, whereas smaller support regions lead to non-uniqueness because the body can be altered away from the support without changing the measure (Brauner et al., 13 Aug 2025).

A related disk classification appears as the base case of the axial theory: for $S(K^{[i]},C_1,\dots,C_{n-i-1};\cdot)$8 and a non-negative zonal Borel measure $S(K^{[i]},C_1,\dots,C_{n-i-1};\cdot)$9, there exists a body of revolution $K$0 not a segment such that

$K$1

if and only if $K$2 is centered and not concentrated on $K$3, and $K$4 is unique up to translation by a multiple of $K$5 (Brauner et al., 13 Aug 2025).

4. Rotational symmetry and explicit reconstruction formulas

Under rotational symmetry, the mixed Christoffel–Minkowski problem admits explicit solution formulas. For the modified $K$6-th area measure $K$7, the rotationally symmetric problem asks for a body of revolution $K$8 such that

$K$9

where $\Omega_1,\dots,\Omega_{n-1}$00 is finite, centered, and $\Omega_1,\dots,\Omega_{n-1}$01-invariant. The main theorem states that there exists a body of revolution $\Omega_1,\dots,\Omega_{n-1}$02 with $\Omega_1,\dots,\Omega_{n-1}$03 such that $\Omega_1,\dots,\Omega_{n-1}$04 if and only if $\Omega_1,\dots,\Omega_{n-1}$05 is not concentrated on the equator

$\Omega_1,\dots,\Omega_{n-1}$06

(Mussnig et al., 15 Aug 2025).

The body is reconstructed from two radial convex functions $\Omega_1,\dots,\Omega_{n-1}$07, obtained by solving mixed Monge–Ampère equations on $\Omega_1,\dots,\Omega_{n-1}$08. The resulting convex body is

$\Omega_1,\dots,\Omega_{n-1}$09

and its support function is given explicitly on the lower hemisphere, equator, and upper hemisphere by formulas involving first moments of $\Omega_1,\dots,\Omega_{n-1}$10 over the spherical caps

$\Omega_1,\dots,\Omega_{n-1}$11

The cap functions

$\Omega_1,\dots,\Omega_{n-1}$12

encode the lower and upper boundary profiles of the body, while the equatorial mass $\Omega_1,\dots,\Omega_{n-1}$13 determines the possible vertical segment in the body (Mussnig et al., 15 Aug 2025).

In the general mixed problem

$\Omega_1,\dots,\Omega_{n-1}$14

with fixed reference bodies of revolution $\Omega_1,\dots,\Omega_{n-1}$15, the same reconstruction scheme persists, but the cap moments are normalized by the profile functions of the reference bodies. The analytic engine is an explicit reduction of radially symmetric mixed Monge–Ampère and Hessian equations to one-dimensional integral equations. For radially symmetric convex $\Omega_1,\dots,\Omega_{n-1}$16, the quantity

$\Omega_1,\dots,\Omega_{n-1}$17

equals the left derivative of the radial profile, and for radially symmetric $\Omega_1,\dots,\Omega_{n-1}$18,

$\Omega_1,\dots,\Omega_{n-1}$19

This gives existence and uniqueness of the radial convex solution up to an additive constant, and then an explicit support-function formula for the body of revolution follows from convex conjugation (Mussnig et al., 15 Aug 2025).

These rotationally symmetric results show that, in high-symmetry regimes, the mixed Christoffel problem is not merely solvable but explicitly invertible. The measure determines cap-mass functions, the cap-mass functions determine radial slopes, and the profile functions integrate to produce the body.

5. Nonlinear Christoffel–Minkowski extensions and curvature flows

A major strand of current work broadens the mixed Christoffel problem from mixed area measures to prescribed mixed curvature measures and fully nonlinear Hessian equations. One direction introduces the $\Omega_1,\dots,\Omega_{n-1}$20-th $\Omega_1,\dots,\Omega_{n-1}$21-mixed curvature measure

$\Omega_1,\dots,\Omega_{n-1}$22

and formulates the $\Omega_1,\dots,\Omega_{n-1}$23-Christoffel–Minkowski problem as the problem of characterizing measures of the form $\Omega_1,\dots,\Omega_{n-1}$24. This framework simultaneously extends the classical Christoffel-Minkowski problem, the $\Omega_1,\dots,\Omega_{n-1}$25 Christoffel-Minkowski problem, and the $\Omega_1,\dots,\Omega_{n-1}$26 dual Minkowski problem. In the smooth strictly convex setting, the authors solve the existence problem by an expanding curvature flow and prove uniqueness under the same inequality $\Omega_1,\dots,\Omega_{n-1}$27 (Chen et al., 2022).

A second flow-based approach studies an anisotropic expanding flow with speed $\Omega_1,\dots,\Omega_{n-1}$28. After normalization, the flow converges smoothly to a support function solving

$\Omega_1,\dots,\Omega_{n-1}$29

For $\Omega_1,\dots,\Omega_{n-1}$30, $\Omega_1,\dots,\Omega_{n-1}$31, and a positive smooth even $\Omega_1,\dots,\Omega_{n-1}$32 satisfying the paper’s convexity condition, the normalized flow exists for all time and converges to the unique smooth solution of the $\Omega_1,\dots,\Omega_{n-1}$33 Christoffel-Minkowski problem. The authors emphasize that this yields a new parabolic proof of Guan–Xia’s existence theorem for $\Omega_1,\dots,\Omega_{n-1}$34, avoiding the constant rank theorem (Zhang, 2023).

A further mixed $\Omega_1,\dots,\Omega_{n-1}$35 problem prescribes a measure that combines the $\Omega_1,\dots,\Omega_{n-1}$36-th area measure with the $\Omega_1,\dots,\Omega_{n-1}$37-th dual curvature measure. Its governing equation is

$\Omega_1,\dots,\Omega_{n-1}$38

For $\Omega_1,\dots,\Omega_{n-1}$39 and $\Omega_1,\dots,\Omega_{n-1}$40, if $\Omega_1,\dots,\Omega_{n-1}$41 is positive, even, smooth, and satisfies

$\Omega_1,\dots,\Omega_{n-1}$42

then there exists a smooth, origin-symmetric, strictly convex solution. The proof uses a new weighted gradient estimate, a full-rank theorem, and Leray–Schauder degree theory. For $\Omega_1,\dots,\Omega_{n-1}$43 the paper does not claim a general uniqueness theorem, although for the dual Minkowski case $\Omega_1,\dots,\Omega_{n-1}$44 it proves uniqueness near $\Omega_1,\dots,\Omega_{n-1}$45 in $\Omega_1,\dots,\Omega_{n-1}$46 (Cabezas-Moreno et al., 7 Apr 2025).

A more general PDE perspective studies equations of the form

$\Omega_1,\dots,\Omega_{n-1}$47

These equations are described as generalizations of the equations for the Christoffel-Minkowski problem and as prescriptions of a convex combination of area measures. On the sphere they admit admissible $\Omega_1,\dots,\Omega_{n-1}$48-invariant solutions under symmetry assumptions, but admissibility is only in the cone $\Omega_1,\dots,\Omega_{n-1}$49, not necessarily the full convex cone $\Omega_1,\dots,\Omega_{n-1}$50. The paper explicitly notes that a constant rank theorem would be needed to upgrade admissibility to convexity, and that this is missing for the inhomogeneous equation (Guan et al., 2019).

Together, these developments show that the mixed Christoffel problem now functions both as a concrete inverse problem for mixed area measures and as a template for a family of nonlinear prescribed-curvature problems.

6. Boundary and non-Euclidean analogues

The mixed Christoffel philosophy has been extended to capillary geometry in the Euclidean half-space. In this setting one replaces the full sphere by a spherical cap and classical area measures by capillary area measures. For a strictly convex capillary hypersurface, the support function $\Omega_1,\dots,\Omega_{n-1}$51 on the cap $\Omega_1,\dots,\Omega_{n-1}$52 satisfies the Hessian equation

$\Omega_1,\dots,\Omega_{n-1}$53

where the second line is a Robin boundary condition encoding the contact angle. A capillary Christoffel–Minkowski problem is then formulated by prescribing the $\Omega_1,\dots,\Omega_{n-1}$54-th capillary area measure

$\Omega_1,\dots,\Omega_{n-1}$55

For $\Omega_1,\dots,\Omega_{n-1}$56 and $\Omega_1,\dots,\Omega_{n-1}$57, existence and uniqueness up to horizontal translation are proved under the condition that $\Omega_1,\dots,\Omega_{n-1}$58 is connected to $\Omega_1,\dots,\Omega_{n-1}$59 in the class $\Omega_1,\dots,\Omega_{n-1}$60, which packages an orthogonality condition and convexity of $\Omega_1,\dots,\Omega_{n-1}$61 together with a boundary monotonicity requirement (Mei et al., 18 Dec 2025).

A closely related capillary result gives the half-space analogue of the Guan–Ma theorem. If $\Omega_1,\dots,\Omega_{n-1}$62 satisfies the horizontal moment conditions

$\Omega_1,\dots,\Omega_{n-1}$63

together with

$\Omega_1,\dots,\Omega_{n-1}$64

then there exists a smooth strictly convex capillary hypersurface whose capillary support function $\Omega_1,\dots,\Omega_{n-1}$65 solves

$\Omega_1,\dots,\Omega_{n-1}$66

and the hypersurface is unique up to horizontal translation. This is presented explicitly as the capillary counterpart of the mixed Christoffel philosophy, with the Robin boundary condition as the essential new feature (Hu et al., 12 Apr 2025).

An older non-Euclidean analogue arises in Lorentzian geometry. There the ambient objects are $\Omega_1,\dots,\Omega_{n-1}$67-convex sets in Minkowski space, the parameter space is $\Omega_1,\dots,\Omega_{n-1}$68, and the first area measure satisfies

$\Omega_1,\dots,\Omega_{n-1}$69

in the smooth case. The Lorentzian Christoffel problem asks for necessary and sufficient conditions for a positive Radon measure on $\Omega_1,\dots,\Omega_{n-1}$70 to be the first area measure of an $\Omega_1,\dots,\Omega_{n-1}$71-convex set. The paper also develops a mixed area-measure formalism via mixed covolume in the Fuchsian setting, and states that the Christoffel problem is completely solved there (Fillastre et al., 2013).

These analogues preserve the central structure of the mixed Christoffel problem—support-function parametrization, curvature radii encoded by a Hessian-type tensor, translation or horizontal-translation orthogonality, and an inverse problem for area measures—while changing the ambient geometry, the domain of the support function, and, in the capillary case, the boundary conditions.