### Non-Technical Analogy

The mathematician Stephen Schanuel once asked the question "What is the length of a potato?"
The question sounds strange—potatoes are irregularly shaped 3-dimensional objects, and it seems difficult to quantify the "length" of a potato.
*Length* generally makes sense for lines and paths, but Schanuel wanted a concept of length for other objects.
Indeed, Schanuel gave a sensible answer to his question, describing a mathematical way of understanding the the "length" of any object.

I answered an analogous question in my Ph.D. thesis: "What is the temperature of a potato?"
An object, such as a potato, is usually not the same temperature at every point in its interior, making it difficult to say what is *the* temperature of a potato.
Yet, if we know the temperature at all points inside the potato, can we quantify its average temperature?
To use a bit of mathematical terminology, we can think of the temperature of a potato as a *function* defined at all points in the potato.
The question I answered in my thesis is then "What is the size of a function?"
I showed that there are various ways of answering this question, and I proved a theorem classifying notions of *size* for functions.

### A Bit More Detail

A *valuation* is a way of assigning a notion of size to sets.
Hugo Hadwiger classified all valuations (under reasonable assumptions) on compact convex sets:
Hadwiger's Theorem says that all such valuations comprise a vector space spanned by the intrinsic volumes.
For convex sets in **R**^{n}, there are *n* + 1 intrinsic volumes, which generalize both Lebesgue measure and Euler characteristic.
Euler characteristic can be considered a topological notion of size; the other intrinsic volumes are geometric notions of size.

We can generalize the concept of valuation from sets to *functions* defined on sets.
As the intrinsic volumes are valuations on sets, the *Hadwiger integrals* are valuations on real-valued functions.
We can think of the Hadwiger integrals as topological and geometric integrals — or alternately, as topological and geometric notions of size for functions.
I have proved a Hadwiger theorem classifying valuations on functions in terms of the Hadwiger integrals.
The full theorem requires careful consideration of continuity and topology, and the details are available in my paper.
I am now exploring valuations in other contexts, such as simplicial maps, as well as their real-world applications.

### Want to read more?

- Stephen Schanuel's paper,
*What is the length of a potato?*, is an enjoyable and accessible introduction to the concept of valuations. *Hadwiger's Theorem for Definable Functions*—my recent paper with Robert Ghrist and Yuliy Baryshnikov, is now published in Advances in Mathematics.- See my list of papers.