---
tag: S34G7679AK
kind: definition
name: Polyconvexity
msc: [49J45, 26B25, 74B20]
defines: 8VYYWQ5CMB
uses: [07GR3TNW4H, 0BT9BYWVRY, DVJ17KVWXP, GK4P8G0PB2]
sources:
  - url: https://en.wikipedia.org/wiki/Polyconvex_function
---

## Definition

Let $m, n \ge 1$ and let $T(\xi) \in \R^{\tau}$ denote the vector of all [[minors|0BT9BYWVRY]] of all orders $1, \dots, \min(m, n)$ of a matrix $\xi \in \R^{m \times n}$.
A function $f \colon \R^{m \times n} \to (-\infty, +\infty]$ is *polyconvex* if there is a [[convex function|07GR3TNW4H]] $g \colon \R^{\tau} \to (-\infty, +\infty]$ with
$$f(\xi) = g\big(T(\xi)\big) \quad \text{for all } \xi \in \R^{m \times n}.$$

## Remarks

For $m = n = 2$, $T(\xi) = (\xi, \det \xi) \in \R^5$.
For $m = n = 3$, $T(\xi) = (\xi, \operatorname{cof} \xi, \det \xi) \in \R^{19}$, with $\operatorname{cof} \xi$ the cofactor matrix.

Example: for $m = n = 2$, $f(\xi) = \det \xi$ is polyconvex, with $g$ linear, although $f$ is not convex.

Example: $f(\xi) = \abs{\xi}^2 + h(\det \xi)$ for $3 \times 3$ matrices, with $h(d) = d^{-1}$ for $d > 0$ and $h(d) = +\infty$ for $d \le 0$.
It is polyconvex, since $h$ is convex on $\R$ with values in $(-\infty, +\infty]$.

Non-example: for $m = n = 2$, $f(\xi) = -(\det \xi)^2$ is not polyconvex.
Along $\xi(t) = I + t \, e_1 \otimes e_1$ it equals $-(1 + t)^2$, which is not convex.
So $f$ is not [[rank-one convex|GK4P8G0PB2]], while every finite polyconvex function is.

For real-valued $f$, polyconvexity implies [[quasiconvexity|DVJ17KVWXP]], and the converse fails when $m, n \ge 2$.