 Non-identifiability of shape from height and mass

Non-identifiability of shape from height and mass means that different external geometries can produce the same measured height and mass. The observation map from a body shape to those two numbers is many-to-one. A formula using only those inputs cannot distinguish shapes that the observations themselves do not distinguish. This is an identifiability issue in an inverse problem, independently of measurement precision.

An original family of homogeneous cuboids supplies an explicit proof. Fix height $H$, mass $M$ and density $\rho$, and define horizontal area $A=M/(\rho H)$. LetEvery real $t$ gives $ab=A$, the same height and the same mass. Its external surface area, however, isThe square section $t=0$ minimizes this family's area. At $t=\log 2$, the sides have ratio four and the lateral surface area is 25% larger. Height and mass remain exactly unchanged. This is a geometric construction, not a proposed anatomical model of a person.

With stipulated $H=1.8$ m, $M=80$ kg and $\rho =1000\mathrm{kg}{\mathrm{m}}^{-3}$, those two total areas are approximately $1.607$ and $1.986{\mathrm{m}}^2$. The Mosteller formula assigns both the same predicted body surface area, $2.00{\mathrm{m}}^2$, because its documented inputs are only height and mass.^[ref] No reduction of numerical rounding can make one identical prediction equal both different geometric areas.

Additional measurements and anatomical identifiability can change the problem. In this particular cuboid family, measuring the horizontal perimeter $P=2(a+b)$ identifies the area through $S=2A+HP$. That conclusion relies on the assumed geometry; a perimeter does not generally determine the full external surface of an arbitrary human body.

Validation must therefore ask whether the observations and model contain the shape information needed for the intended quantity. Repeated precise height and mass measurements constrain the inputs. Independent geometric measurements or a validated restriction on allowable shapes constrain the missing anatomical information.