1 Introduction
Bilogconcavity (of a probability measure on the real line) is a property recently introduced by Dümbgen, Kolesnyk and Wilke ([DKW17]), that aims at bypassing some restrictive aspects of logconcavity while preserving some of its nice features. More precisely, bilogconcavity amounts to logconcavity of both and and a simple application of Prékopa’s theorem on stability of logconcavity through marginalization ([Pré73], see also [SW14] for a discussion on the various proofs of this fundamental theorem) shows that logconcave measures are also bilogconcave (see [BB05] for a more direct, elementary proof of this latter fact).
From a modelisation perspective, bilogconcavity and logconcavity may be seen as shape constraints. In statistics, when they are available, shape constraints represent an interesting alternative to more classical parametric, semiparametric or nonparametric approaches and constitute an active contemporary line of research ([Wal09, Sam18]). Bilogconcavity was indeed proposed in the aim to contribute to this research area ([DKW17]). It was used in [DKW17]
to construct efficient confidence bands for the cumulative distribution function and some functionals of it. The authors highlight that bilogconcave measures admit multimodal measures while it is wellknown that logconcave measures are unimodal. Furthermore, Dümbgen et al.
[DKW17] establish the following characterization of bilogconcave distributions. For a distribution function , denoteand call “nondegenerate”, the functions such that .
Theorem 1 (Characterization of bilogconcavity, [Dkw17])
Let be a nondegenerate distribution function. The following four statements are equivalent:
 (i)

is bilogconcave, i.e. and are logconcave functions in the sense that their logarithm is concave.
 (ii)

is continuous on and differentiable on with derivative such that, for all and ,
 (iii)

is continuous on and differentiable on with derivative such that the hazard function is nondecreasing and reverse hazard function is nonincreasing on .
 (iv)

is continuous on and differentiable on with bounded and strictly positive derivative . Furthermore, is locally Lipschitz continuous on with derivative satisfying
Note that if one includes degenerate measures  that is Dirac masses  it is easily seen that the set of bilogconcave measures is closed under weak limits.
Just as concave measures generalize logconcave ones, Laha and Wellner [LW17] proposed the concept of biconcavity, that generalize bilogconcavity and that include concave densities. Some characterizations of biconcavity, that extend the previous theorem, are derived in [LW17].
On the probabilistic side, even if some characterizations are available, many important questions remain about the properties of bilogconcave measures. Indeed, logconcave measures satisfy many nice properties (see for instance [Gué12, SW14, Col17] and references therein) and it is natural to ask whether some of those are extended to bilogconcave measures. Answering this question is the primary object of this note.
We show in Section 2 that the isoperimetric constant of a bilogconcave measure is simply equal to two times the value of its density with respect to the Lebesgue measure  that indeed exists  at its median, thus extending a property available for logconcave measures. We deduce that a bilogconcave measure has exponential tails, also extending a property valid in the logconcave case.
In Section 3, we show that the convolution of a logconcave measure and a bilogconcave measure is bilogconcave. As a consequence, we get that any bilogconcave measure can be approximated by a sequence of bilogconcave measures having regular densities. Furthermore, we give a necessary and sufficient condition for the convolution of two bilogconcave measures to be bilogconcave.
Finally, we discuss in Section 3.1 possible ways to obtain a multivariate notion of bilogconcavity. This problem is not a priori obvious, because the definition of bilogconcavity in one dimension relies on the cumulative distribution function and so, on the total order existing on real numbers. To this end, we derive a characterization of (symmetric) bilogconcave measures on
through their isoperimetric profile. Then we propose a multidimensional generalization for symmetric measures by considering their isoperimetric profile, restricted to half spaces. We conclude by discussing a way to strengthen the latter definition in order to ensure stability through convolution by any logconcave measure. The question of providing a nice definition of bilogconcavity in higher dimension, that would also impose existence of some exponential moments, remains open.
2 Isoperimetry and concentration for bilogconcave measures
Let be the distribution function of a probability measure on the real line. Assume that is nondegenerate (in the sense of its distribution function being nondegenerate) and let be the density of its absolutely continuous part.
Recall the following formula for the isoperimetric constant of , due to Bobkov and Houdré [BH97],
The following theorem extends a wellknown fact related to isoperimetric constant for logconcave measure to the case of bilogconcave measures.
Theorem 2
Let be a probability measure with nondegenerate distribution function being bilogconcave. Then admits a density on and it holds
where is the median of .
In general, the isoperimetric constant is hard to compute, but in the bilogconcave case Theorem 2 provides a straightforward formula, that extends a formula valid for logconcave measures (see for instance [SW14]).
In the following, we will also use the notation .
Proof. Note that the median is indeed unique by Theorem 1 above. For ,
As is bilogconcave, is thus nonincreasing on . For ,
Thus, is nondecreasing on . Consequently, the maximum of is attained on and its value is .
Corollary 3
Let as above be a bilogconcave measure with median . Then and satisfies the following Poincaré inequality: for any square integrable function with derivative ,
(1) 
where
is the variance of
with respect to . Consequently, has bounded Orlicz norm and achieves the following exponential concentration inequality,(2) 
where is the concentration function of , defined by , where and is the (open) neighborhood of .
As it is wellknown (see [Led01] for instance), inequality (2) implies that for any Lipschitz function ,
where is a median of , that is and .
Proof. The fact that is given by point (iii) of Theorem 1 above. Then Inequality (1) is a consequence of Theorem 2
via Cheeger’s inequality for the first eigenvalue of the Laplacian (see for instance Inequality 3.1 in
[Led01]). Inequality (2) is a classical consequence of Inequality (1) as well (see Theorem 3.1 in [Led01]).Note that, following Bobkov [Bob96], for a logconcave probability measure on having a positive density on , the function is concave. By Theorem 1 above, bilogconcavity of reduces to nonincreasingness of the functions and , which is equivalent to nonincreasingness of and . As is concave for a logmeasure and as , bilogconcavity follows. This gives another proof of the fact that logconcave measures are bilogconcave.
Example 4
The function is in general hard to compute. But a few easy examples exist. For instance, for the logistic distribution, , we have . For the Laplace distribution, , .
3 Stability through convolution
Take and
two independent random variables with respective distribution functions
and that are bilogconcave. Hence and have densities, denoted by and . Then(3) 
In addition,
(4) 
Proposition 5
If is bilogconcave, is logconcave and is independent from , then is bilogconcave.
Proof. By using formulas (3) and (4), this is a direct application of Prékopa’s theorem ([Pré73]) on the marginal of a logconcave function.
Corollary 6
Take a (nondegenerate) bilogconcave measure on , with density . Then there exists a sequence of infinitely differentiable bilogconcave densities, positive on , that converge to in , for any
Corollary 6 is also an extension of an approximation result available in the set of logconcave distributions, see [SW14, Section 5.2].
Proof. It suffices to consider the convolution of with a sequence of centered Gaussian densities with variances converging to zero. As has an exponential moment, it belongs to any , Then a simple application of classical theorems about convolution in (see for instance [Rud87, p. 148]) allows to check that the approximations converge to in any ,
More generally, the following theorem gives a necessary and sufficient condition for the convolution of two bilogconcave measures to be bilogconcave.
Theorem 7
Take and two independent bilogconcave random variables with respective densities and and cumulative distribution functions and . Denote and and consider for any , the following measures on ,
and
Then is bilogconcave if and only if for any ,
(5) 
and
(6) 
Of course, a simple symmetrization argument shows that conditions (5) and (6) are satisfied if pointwise, which means that is logconcave, in which case we recover Proposition 5 above. But Theorem 7 is more general. Indeed, it is easily checked by direct computations that the convolution the Gaussian mixture  which is bilogconcave but not logconcave, see [DKW17, Section 2]  with itself is bilogconcave.
To prove Theorem 7, we will use the following lemma.
Lemma 8
Take such that and a measure on with density absolutely continuous and . Take Lipschitz continuous such that and
then
Proof of Lemma 8. This a simple integration by parts: from the assumptions, we have
Proof of Theorem 7. Recall that we have
Our first goal is to find some conditions such that is logconcave. It is sufficient to prove that, for any ,
or equivalently,
Denote . We have
Furthermore, we get
Now, by Lemma 8, it holds,
Gathering the equations, we get
which gives condition (5). Likewise condition (6) arises from the same type of computations when studying logconcavity of .
3.1 Towards a multivariate notion of bilogconcavity
Let us introduce this section with the following remark. The isoperimetric profile is defined as follows: for any ,
where , with . Note that the isoperimetric profile depends on the distance that is considered. Unless explicitly mentioned, we will consider in the following that the distance is the Euclidean distance. From inequality (2.1) in [Bob96], we have for a logconcave measure on and any ,
Hence,
If is moreover symmetric, then for any .
For a general measure , we define the isoperimetric profile restricted to halfspaces : for any ,
For a measure on having a density , one has
since possible halfspaces are in this case or . If is symmetric, then and bilogconcavity is equivalent to nonincreasingness of on . As proved in Bobkov [Bob96, Proposition A.1], logconcavity on is actually equivalent to concavity of .
Furthermore, as previously remarked, in the onedimensional logconcave case. The latter identity is still true in higher dimension for the Gaussian measure when the distance is given by the Euclidean norm (see [Bor75]) and this characterizes Gaussian measures. In general, it also holds pointwise.
Take the distance to be given by the supnorm, . Then, for any set , we have . In this case, Bobkov [Bob96, Theorem 1.1] characterizes symmetric logconcave measures for which .
Reverse relation in higher dimension between and when
is logconcave, is related to the socalled KLShyperplane conjecture (see for instance
[LV17, LV18]).A possible extension of the notion of bilogconcavity is the following.
Definition 9
Let be a probability measure on . Assume that is symmetric around the origin. Then is said to be weakly bilogconcave (with respect to the distance ) if the function
is nonincreasing on .
The latter definition extends the definition of bilogconcavity for symmetric measures on the real line. However, we consider that the definition is “weak” since, as we will see, it seems in fact natural to ask for more. In the following, a symmetric measure is a measure that is symmetric around the origin.
Proposition 10
Symmetric logconcave measures on are bilogconcave (for the Euclidean distance).
Proof. Take
a unit vector and consider the measure
defined to be the projection of the measure on the line containing and directed by . Consequently, is a logconcave measure on , symmetric around zero. Hence is concave and consequently, is nonincreasing. Since halfspaces are parameterized by unitary vectors together with a point on the line containing zero and directed by the considered unitary vector, this readily gives the nonincreasingness of .One can notice that the latter proof is in fact only based on stability of logconcavity through onedimensional marginalizations. This naturally leads to the following second definition of bilogconcavity in higher dimension.
Definition 11
Let be a probability measure on . Then is said to be weakly bilogconcave if for every line , the (Euclidean) projection measure of onto the line is a (onedimensional) bilogconcave measure on (that can be possibly degenerate). More explicitly, for any and any Borel set ,
where is a unit directional vector of the line .
Note that weakly bilogconcave measures are not necessarily symmetric. In the case of symmetric measures, the notion of weakly bilogconcavity is actually a strengthening of Definition 9.
Proposition 12
Let be a symmetric, weakly bilogconcave probability measure on . Then is weakly bilogconcave.
Proof. By parametrization of halfspaces, we have the following formula, for any ,
Then the conclusion follows by noticing that for any line such that , the projection measure of is symmetric and bilogconcave. Hence, is nonincreasing on and so is .
The following result states that the notion of weakly bilogconcavity is stable through convolution by logconcave measures.
Proposition 13
The convolution of a logconcave measure with a weakly bilogconcave one is weakly bilogconcave.
Proof. The formula shows that the projection of the convolution of two measures on a line is the convolution of the projections of measures on this line. This allows to reduce the stability through convolution by a logconcave measure to dimension one and concludes the proof.
As for the logconcave case, it is moreover directly seen that weakly and weakly logconcavity are stable by affine transformations of the space.
Actually, in addition to containing logconcave measures and being stable through convolution by a logconcave measure, there are at least two other properties that one would naturally require for a convenient multidimensional concept of bilogconcavity: existence of a density with respect to the Lebesgue measure on the convex hull of its support and existence of a finite exponential moment for the (Euclidean) norm. We can express this latter remark through the following open problem, that concludes this note.
Open Problem: Find a nice characterization of probability measures on that are weakly bilogconcave, that admit a density with respect to the Lebesgue measure on the convex hull of their support and whose Euclidean norm has exponentially decreasing tails.
Acknowledgement 14
I express my deepest gratitude to Jon Wellner, who introduced me to the intriguing notion of bilogconcavity and who provided several numerical computations during his visit at the CrestEnsai, that helped to understand the convolution problem for bilogconcave measures. Many thanks also to Jon for his comments on a previous version of this note.
References
 [BB05] M. Bagnoli and T. Bergstrom, Logconcave probability and its applications, Econom. Theory 26 (2005), no. 2, 445–469. MR MR2213177
 [BH97] S. G. Bobkov and C. Houdré, Isoperimetric constants for product probability measures, Ann. Probab. 25 (1997), no. 1, 184–205. MR 1428505
 [Bob96] S. Bobkov, Extremal properties of halfspaces for logconcave distributions, Ann. Probab. 24 (1996), no. 1, 35–48. MR 1387625 (97e:60027)
 [Bor75] C. Borell, The BrunnMinkowski inequality in Gauss space, Invent. Math. 30 (1975), no. 2, 207–216. MR 0399402
 [Col17] A. Colesanti, Logconcave functions, Convexity and concentration, IMA Vol. Math. Appl., vol. 161, Springer, New York, 2017, pp. 487–524. MR 3837280
 [DKW17] L. Dümbgen, P. Kolesnyk, and R. A. Wilke, Bilogconcave distribution functions, J. Statist. Plann. Inference 184 (2017), 1–17. MR 3600702
 [Gué12] O. Guédon, Concentration phenomena in high dimensional geometry, Proceedings of the Journées MAS 2012 (ClermondFerrand, France), 2012.
 [Led01] M. Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs, vol. 89, American Mathematical Society, Providence, RI, 2001. MR 1849347 (2003k:28019)
 [LV17] Y. T. Lee and S. S. Vempala, Eldan’s stochastic localization and the KLS hyperplane conjecture: An improved lower bound for expansion, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 1517, 2017, 2017, pp. 998–1007.
 [LV18] , The KannanLovászSimonovits conjecture, CoRR abs/1807.03465 (2018).
 [LW17] N. Laha and J. A. Wellner, Bioncave istributions, arXiv:1705.00252, preprint.
 [Pré73] A. Prékopa, On logarithmic concave measures and functions, Acta Sci. Math. (Szeged) 34 (1973), 335–343. MR 0404557 (53 #8357)
 [Rud87] W. Rudin, Real and complex analysis, third ed., McGrawHill Book Co., New York, 1987. MR 924157 (88k:00002)

[Sam18]
R. J. Samworth,
Recent progress in logconcave density estimation
, Statist. Sci. 33 (2018), no. 4, 493–509. MR 3881205  [SW14] A. Saumard and J. A. Wellner, Logconcavity and strong logconcavity: A review, Statist. Surv. 8 (2014), 45–114.
 [Wal09] G. Walther, Inference and modeling with logconcave distributions, Statist. Sci. 24 (2009), no. 3, 319–327. MR 2757433 (2011j:62110)
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