**1. Introduction**

The isoperimetric inequality is an important inequality in geometry which originated from the well-known isoperimetric problem. The isoperimetric inequality has a profound influence on each branches of mathematics. The breakthrough works of Federer and Fleming [1] and Mazya [2] discovered independently the connection between the isoperimetric problem and the Sobolev embedding problem. They established the sharp Sobolev inequality by using the isoperimetric inequality. This exciting connection has motivated a number of studies in recent years about interactions of geometric and analytic inequalities. In this paper, we further study the connection between geometry and analysis.

Let us recall some facts about convex bodies. Let *K* be a convex body (i.e., compact, convex subset with non-empty interior) in the *n*-dimensional Euclidean space R*<sup>n</sup>*, the family {*TK* : *T* ∈ SL(n)} of its positions are studied by many mathematicians. Introducing the right position of the unit ball *KX* of a finite dimensional normed space *X* is one of the main problems in the asymptotic theory. There exist many celebrated positions for different purposes, for example isotropic position, *M*-position, John's position, the -position and so on, see [3,4].

Our purpose is to study the isotropic position of log-concave functions. Hence, we first recall some geometric backgrounds and these are our motivations. Let *K* be a convex body in R*n* with centroid at the origin and volume equal to one. A convex body *K* is in isotropic position if

$$\int\_{K} \langle \mathfrak{x}, \theta \rangle^2 d\mathfrak{x} = L\_{K'}^2 \,\,\forall \theta \in \mathbb{S}^{n-1}.$$

where ·, · is the usual inner product in R*n* and *Sn*−<sup>1</sup> is the unit sphere in R*<sup>n</sup>*. It's worth noting that every convex body *K* with volume one has an isotropic position, and this position is uniqueness (up to an orthogonal transformation), see, e.g., [3]. Isotropic positions have been used to study the classical convexity problems, for example, the minimal surface area of a convex body and its extension [5,6], the minimal mean width of a convex body and its extension [7,8]. Other contributions include e.g., [9–11] among others.

We recall two specific examples on isotropic positions. Let *K* be a convex body and denote by *S*(*K*) its surface area. If *S*(*K*) ≤ *S*(*TK*) for every *T* ∈ SL(n), then *K* has minimal surface area (see, e.g., [5]). Petty [5] obtained the following characterization of the minimal surface area position: a convex body *K* has minimal surface area if and only if its surface area measure *σK* is isotropic, i.e.,

$$\int\_{S^{n-1}} \langle u, \theta \rangle^2 d\sigma \chi(u) = \frac{S(K)}{n}, \quad \forall \theta \in S^{n-1}.$$

As a second example, the minimal mean width will be recalled which was defined by Giannopoulos and Milman [7]. Let *K* be a convex body in R*<sup>n</sup>*, the mean width *w*(*K*) of *K* is define as

$$w(\boldsymbol{k}) = 2 \int\_{S^{n-1}} h\_{\mathcal{K}}(\boldsymbol{u}) d\sigma(\boldsymbol{u}),$$

where *hK*(*u*) := sup*y*∈*<sup>K</sup><sup>u</sup>*, *y* is the support function of *K* and *σ* is the rotationally invariant probability measure on *Sn*−1. For every *T* ∈ SL(n), if *w*(*K*) ≤ *w*(*TK*) then *K* has minimal mean width (see, e.g., [7]). Giannopoulos and Milman [7] showed that if the support function of *K* is twice continuously differentiable, then *K* has minimal mean width if and only if the measure *d<sup>ν</sup>K* = *hKdσ* is isotropic, i.e.,

$$\int\_{S^{n-1}} h\_K(u) \langle u, \theta \rangle^2 d\sigma(u) = \frac{w(K)}{2n}, \quad \forall \theta \in \mathcal{S}^{n-1}.$$

Within the last few years, many geometric results have been generalized to their corresponding functional versions, including but not limited to the functional version Blaschke-Santaló inequality and its reverse [12–16], the functional affine surface areas [17–19], Minkowski problem for functions [20–22], and analytic inequalities with geometric background [23–28].

In this paper, we consider the log-concave functions in R*<sup>n</sup>*. A function *f* : R*n* → R is log-concave if for any *x*, *y* ∈ R*n* and *λ* ∈ [0, 1], it holds

$$f(\lambda x + (1 - \lambda)y) \ge f(x)^{\lambda} f(y)^{1 - \lambda}. \tag{1}$$

A typical example of log-concave functions is the characteristic function of convex bodies, **1***K* (which is defined as **<sup>1</sup>***K*(*x*) = 1 when *x* ∈ *K* and **<sup>1</sup>***K*(*x*) = 0 when *x* ∈ *K*). Let *J*(*f*) denote the total mass functional of *f* : R*n* → R, namely

$$I(f) = \int\_{\mathbb{R}^n} f(x)dx.$$

For any *t* > 0 and log-concave functions *f* , *g* : R*n* → R, Colesanti and Fragala` [21] defined the first variation of *J* at *f* along *g* as

$$\delta J(f, \mathbf{g}) = \lim\_{t \to 0^{+}} \frac{J(f \oplus t \cdot \mathbf{g}) - J(f)}{t},\tag{2}$$

where *t* · *g*(*x*) = *g<sup>t</sup>*(*x*/*t*) for *t* > 0 and *x* ∈ R*<sup>n</sup>*, and *f* ⊕ *g* the Asplund sum of functions *f* and *g*, i.e.,

$$[f \oplus g](\mathbf{x}) = \sup\_{\mathbf{x} = \mathbf{x}\_1 + \mathbf{x}\_2} f(\mathbf{x}\_1)g(\mathbf{x}\_2), \quad \mathbf{x} \in \mathbb{R}^n. \tag{3}$$

It was proved that if *f* and *g* are restricted to a subclass of log-concave functions, then the first variation *δJ*(*f* , *g*) precisely turns out to be *Lp* mixed volume of convex bodies (see Proposition 3.13 in [21]). In particular, the perimeter of *f* is defined as (see [21])

$$P(f) = \delta f(f, \gamma\_n)\_{\prime\prime}$$

where *<sup>γ</sup>n*(*x*) = *e*<sup>−</sup> *x*-2 2 is the Gaussian function and *x*- is the Euclidean norm of *x* ∈ R*<sup>n</sup>*.

Motivated by the work of Giannopoulos and Milman [7], we consider the extremal problems of log-concave functions instead of convex bodies, and our purpose is to discuss the possibility of an isometric approach to these questions. We introduce the notion of minimal perimeters of log-concave functions. Assume that *f* is a log-concave function, we call *f* has minimal perimeter if *P*(*f*) ≤ *P*(*f* ◦ *T*) for every *T* ∈ SL(n). Furthermore, we derive the following characteristic theorem of the minimal perimeter.

**Theorem 1.** *If f* : R*n* → [0, ∞) *is a log-concave function, then f has minimal perimeter if and only if*

$$\frac{\operatorname{tr}(T)}{n}P(f) = \frac{1}{2}\int\_{\mathbb{R}^n} \langle \mathbf{x}, T\mathbf{x} \rangle d\mu\_f(\mathbf{x})\tag{4}$$

*for every T* ∈ GL(n)*. Here tr*(*T*) *denotes the trace of T, and μf* = (∇*u*)(*f* H*n*) *is a Borel measure on* R*n (where* H*n is the n-dimensional Hausdorff measure and u* = − log *f ).*

Theorem 1 implies that the log-concave function *f* has minimal perimeter if and only if *<sup>μ</sup>f*(·) is isotropic, and provides a further example of the connections between the theory of convex bodies and that of functions.

We remark that our works belong to the asymptotic theory of log-concave functions which parallel to that of convex bodies. From a geometric and analytic view of point to study convex bodies is the asymptotic theory of convex bodies which emphasize the dependence of various parameters on the dimension. Isotropic positions for convex bodies play important roles in the asymptotic theory of convex bodies. We are not aware of the related results for log-concave functions. Hence, our work in this paper presents a new connection between convex bodies and log-concave functions and it also leads to a new topic in the study of geometry of log-concave functions. We hope that our work provides some useful tools or ideas in the development of geometry of log-concave functions.
